↑

Properties

Label 24-3e72-1.1-c1e12-0-16
Degree 2424
Conductor 2.253×10342.253\times 10^{34}
Sign 11
Analytic cond. 1.51375×1091.51375\times 10^{9}
Root an. cond. 2.412692.41269
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 9·4-s + 6·5-s + 18·8-s + 18·10-s + 12·11-s + 36·16-s − 18·17-s + 6·19-s + 54·20-s + 36·22-s + 15·23-s + 36·25-s + 12·29-s + 57·32-s − 54·34-s + 6·37-s + 18·38-s + 108·40-s + 15·41-s + 108·44-s + 45·46-s + 21·47-s + 27·49-s + 108·50-s − 18·53-s + 72·55-s + ⋯
L(s)  = 1  + 2.12·2-s + 9/2·4-s + 2.68·5-s + 6.36·8-s + 5.69·10-s + 3.61·11-s + 9·16-s − 4.36·17-s + 1.37·19-s + 12.0·20-s + 7.67·22-s + 3.12·23-s + 36/5·25-s + 2.22·29-s + 10.0·32-s − 9.26·34-s + 0.986·37-s + 2.91·38-s + 17.0·40-s + 2.34·41-s + 16.2·44-s + 6.63·46-s + 3.06·47-s + 27/7·49-s + 15.2·50-s − 2.47·53-s + 9.70·55-s + ⋯

Functional equation

Λ(s)=((372)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((372)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 3723^{72}
Sign: 11
Analytic conductor: 1.51375×1091.51375\times 10^{9}
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 372, ( :[1/2]12), 1)(24,\ 3^{72} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 628.4066841628.4066841
L(12)L(\frac12) ≈\approx 628.4066841628.4066841
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
good2 (1−3pT+9pT2−9p2T3+27pT4−69T5+91T6−69pT7+27p3T8−9p5T9+9p5T10−3p6T11+p6T12)(1+3T−9T3−9T4+3p2T5+37T6+3p3T7−9p2T8−9p3T9+3p5T11+p6T12) ( 1 - 3 p T + 9 p T^{2} - 9 p^{2} T^{3} + 27 p T^{4} - 69 T^{5} + 91 T^{6} - 69 p T^{7} + 27 p^{3} T^{8} - 9 p^{5} T^{9} + 9 p^{5} T^{10} - 3 p^{6} T^{11} + p^{6} T^{12} )( 1 + 3 T - 9 T^{3} - 9 T^{4} + 3 p^{2} T^{5} + 37 T^{6} + 3 p^{3} T^{7} - 9 p^{2} T^{8} - 9 p^{3} T^{9} + 3 p^{5} T^{11} + p^{6} T^{12} )
5 1−6T+36T3+99T4−357T5−952T6+1827T7+1386pT8−4347T9−48204T10+16911T11+218649T12+16911pT13−48204p2T14−4347p3T15+1386p5T16+1827p5T17−952p6T18−357p7T19+99p8T20+36p9T21−6p11T23+p12T24 1 - 6 T + 36 T^{3} + 99 T^{4} - 357 T^{5} - 952 T^{6} + 1827 T^{7} + 1386 p T^{8} - 4347 T^{9} - 48204 T^{10} + 16911 T^{11} + 218649 T^{12} + 16911 p T^{13} - 48204 p^{2} T^{14} - 4347 p^{3} T^{15} + 1386 p^{5} T^{16} + 1827 p^{5} T^{17} - 952 p^{6} T^{18} - 357 p^{7} T^{19} + 99 p^{8} T^{20} + 36 p^{9} T^{21} - 6 p^{11} T^{23} + p^{12} T^{24}
7 1−27T2+22T3+54pT4−513T5−3482T6+6291T7+24327T8−44480T9−136944T10+135972T11+830503T12+135972pT13−136944p2T14−44480p3T15+24327p4T16+6291p5T17−3482p6T18−513p7T19+54p9T20+22p9T21−27p10T22+p12T24 1 - 27 T^{2} + 22 T^{3} + 54 p T^{4} - 513 T^{5} - 3482 T^{6} + 6291 T^{7} + 24327 T^{8} - 44480 T^{9} - 136944 T^{10} + 135972 T^{11} + 830503 T^{12} + 135972 p T^{13} - 136944 p^{2} T^{14} - 44480 p^{3} T^{15} + 24327 p^{4} T^{16} + 6291 p^{5} T^{17} - 3482 p^{6} T^{18} - 513 p^{7} T^{19} + 54 p^{9} T^{20} + 22 p^{9} T^{21} - 27 p^{10} T^{22} + p^{12} T^{24}
11 1−12T+27T2+108T3+477T4−6807T5+5438T6+35433T7+267453T8−1206441T9−948924T10−2888901T11+56650647T12−2888901pT13−948924p2T14−1206441p3T15+267453p4T16+35433p5T17+5438p6T18−6807p7T19+477p8T20+108p9T21+27p10T22−12p11T23+p12T24 1 - 12 T + 27 T^{2} + 108 T^{3} + 477 T^{4} - 6807 T^{5} + 5438 T^{6} + 35433 T^{7} + 267453 T^{8} - 1206441 T^{9} - 948924 T^{10} - 2888901 T^{11} + 56650647 T^{12} - 2888901 p T^{13} - 948924 p^{2} T^{14} - 1206441 p^{3} T^{15} + 267453 p^{4} T^{16} + 35433 p^{5} T^{17} + 5438 p^{6} T^{18} - 6807 p^{7} T^{19} + 477 p^{8} T^{20} + 108 p^{9} T^{21} + 27 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24}
13 1−54T2+4T3+1539T4−243T5−30464T6+10071T7+478116T8−184097T9−508680pT10+1167993T11+87287617T12+1167993pT13−508680p3T14−184097p3T15+478116p4T16+10071p5T17−30464p6T18−243p7T19+1539p8T20+4p9T21−54p10T22+p12T24 1 - 54 T^{2} + 4 T^{3} + 1539 T^{4} - 243 T^{5} - 30464 T^{6} + 10071 T^{7} + 478116 T^{8} - 184097 T^{9} - 508680 p T^{10} + 1167993 T^{11} + 87287617 T^{12} + 1167993 p T^{13} - 508680 p^{3} T^{14} - 184097 p^{3} T^{15} + 478116 p^{4} T^{16} + 10071 p^{5} T^{17} - 30464 p^{6} T^{18} - 243 p^{7} T^{19} + 1539 p^{8} T^{20} + 4 p^{9} T^{21} - 54 p^{10} T^{22} + p^{12} T^{24}
17 (1+9T+111T2+711T3+4893T4+23337T5+112057T6+23337pT7+4893p2T8+711p3T9+111p4T10+9p5T11+p6T12)2 ( 1 + 9 T + 111 T^{2} + 711 T^{3} + 4893 T^{4} + 23337 T^{5} + 112057 T^{6} + 23337 p T^{7} + 4893 p^{2} T^{8} + 711 p^{3} T^{9} + 111 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2}
19 (1−3T+84T2−13pT3+3303T4−8784T5+4137pT6−8784pT7+3303p2T8−13p4T9+84p4T10−3p5T11+p6T12)2 ( 1 - 3 T + 84 T^{2} - 13 p T^{3} + 3303 T^{4} - 8784 T^{5} + 4137 p T^{6} - 8784 p T^{7} + 3303 p^{2} T^{8} - 13 p^{4} T^{9} + 84 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2}
23 1−15T+27T2+18pT3+801T4−21354T5−42640T6+483003T7+3508776T8−14314293T9−94988115T10+69646203T11+2948759967T12+69646203pT13−94988115p2T14−14314293p3T15+3508776p4T16+483003p5T17−42640p6T18−21354p7T19+801p8T20+18p10T21+27p10T22−15p11T23+p12T24 1 - 15 T + 27 T^{2} + 18 p T^{3} + 801 T^{4} - 21354 T^{5} - 42640 T^{6} + 483003 T^{7} + 3508776 T^{8} - 14314293 T^{9} - 94988115 T^{10} + 69646203 T^{11} + 2948759967 T^{12} + 69646203 p T^{13} - 94988115 p^{2} T^{14} - 14314293 p^{3} T^{15} + 3508776 p^{4} T^{16} + 483003 p^{5} T^{17} - 42640 p^{6} T^{18} - 21354 p^{7} T^{19} + 801 p^{8} T^{20} + 18 p^{10} T^{21} + 27 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24}
29 1−12T−27T2+504T3+3204T4−21342T5−148921T6+408474T7+5554593T8−4058532T9−155078586T10+79701552T11+3428546241T12+79701552pT13−155078586p2T14−4058532p3T15+5554593p4T16+408474p5T17−148921p6T18−21342p7T19+3204p8T20+504p9T21−27p10T22−12p11T23+p12T24 1 - 12 T - 27 T^{2} + 504 T^{3} + 3204 T^{4} - 21342 T^{5} - 148921 T^{6} + 408474 T^{7} + 5554593 T^{8} - 4058532 T^{9} - 155078586 T^{10} + 79701552 T^{11} + 3428546241 T^{12} + 79701552 p T^{13} - 155078586 p^{2} T^{14} - 4058532 p^{3} T^{15} + 5554593 p^{4} T^{16} + 408474 p^{5} T^{17} - 148921 p^{6} T^{18} - 21342 p^{7} T^{19} + 3204 p^{8} T^{20} + 504 p^{9} T^{21} - 27 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24}
31 1−135T2+382T3+10314T4−43443T5−449990T6+2709261T7+12356577T8−91141832T9−164934846T10+1306511154T11+1847979163T12+1306511154pT13−164934846p2T14−91141832p3T15+12356577p4T16+2709261p5T17−449990p6T18−43443p7T19+10314p8T20+382p9T21−135p10T22+p12T24 1 - 135 T^{2} + 382 T^{3} + 10314 T^{4} - 43443 T^{5} - 449990 T^{6} + 2709261 T^{7} + 12356577 T^{8} - 91141832 T^{9} - 164934846 T^{10} + 1306511154 T^{11} + 1847979163 T^{12} + 1306511154 p T^{13} - 164934846 p^{2} T^{14} - 91141832 p^{3} T^{15} + 12356577 p^{4} T^{16} + 2709261 p^{5} T^{17} - 449990 p^{6} T^{18} - 43443 p^{7} T^{19} + 10314 p^{8} T^{20} + 382 p^{9} T^{21} - 135 p^{10} T^{22} + p^{12} T^{24}
37 (1−3T+165T2−301T3+12591T4−16749T5+586203T6−16749pT7+12591p2T8−301p3T9+165p4T10−3p5T11+p6T12)2 ( 1 - 3 T + 165 T^{2} - 301 T^{3} + 12591 T^{4} - 16749 T^{5} + 586203 T^{6} - 16749 p T^{7} + 12591 p^{2} T^{8} - 301 p^{3} T^{9} + 165 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2}
41 1−15T−27T2+1170T3+1530T4−58866T5−182689T6+52668pT7+15249888T8−49008051T9−1007393688T10+789854886T11+45702053715T12+789854886pT13−1007393688p2T14−49008051p3T15+15249888p4T16+52668p6T17−182689p6T18−58866p7T19+1530p8T20+1170p9T21−27p10T22−15p11T23+p12T24 1 - 15 T - 27 T^{2} + 1170 T^{3} + 1530 T^{4} - 58866 T^{5} - 182689 T^{6} + 52668 p T^{7} + 15249888 T^{8} - 49008051 T^{9} - 1007393688 T^{10} + 789854886 T^{11} + 45702053715 T^{12} + 789854886 p T^{13} - 1007393688 p^{2} T^{14} - 49008051 p^{3} T^{15} + 15249888 p^{4} T^{16} + 52668 p^{6} T^{17} - 182689 p^{6} T^{18} - 58866 p^{7} T^{19} + 1530 p^{8} T^{20} + 1170 p^{9} T^{21} - 27 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24}
43 1−162T2+346T3+12663T4−42741T5−16529pT6+1785402T7+38571471T8−22355525T9−51653403pT10−123766974T11+109827512425T12−123766974pT13−51653403p3T14−22355525p3T15+38571471p4T16+1785402p5T17−16529p7T18−42741p7T19+12663p8T20+346p9T21−162p10T22+p12T24 1 - 162 T^{2} + 346 T^{3} + 12663 T^{4} - 42741 T^{5} - 16529 p T^{6} + 1785402 T^{7} + 38571471 T^{8} - 22355525 T^{9} - 51653403 p T^{10} - 123766974 T^{11} + 109827512425 T^{12} - 123766974 p T^{13} - 51653403 p^{3} T^{14} - 22355525 p^{3} T^{15} + 38571471 p^{4} T^{16} + 1785402 p^{5} T^{17} - 16529 p^{7} T^{18} - 42741 p^{7} T^{19} + 12663 p^{8} T^{20} + 346 p^{9} T^{21} - 162 p^{10} T^{22} + p^{12} T^{24}
47 1−21T+54T2+1287T3+153T4−120774T5+929T6+5893155T7+18070200T8−283676931T9−994660839T10+2777876802T11+83734836753T12+2777876802pT13−994660839p2T14−283676931p3T15+18070200p4T16+5893155p5T17+929p6T18−120774p7T19+153p8T20+1287p9T21+54p10T22−21p11T23+p12T24 1 - 21 T + 54 T^{2} + 1287 T^{3} + 153 T^{4} - 120774 T^{5} + 929 T^{6} + 5893155 T^{7} + 18070200 T^{8} - 283676931 T^{9} - 994660839 T^{10} + 2777876802 T^{11} + 83734836753 T^{12} + 2777876802 p T^{13} - 994660839 p^{2} T^{14} - 283676931 p^{3} T^{15} + 18070200 p^{4} T^{16} + 5893155 p^{5} T^{17} + 929 p^{6} T^{18} - 120774 p^{7} T^{19} + 153 p^{8} T^{20} + 1287 p^{9} T^{21} + 54 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24}
53 (1+9T+210T2+1872T3+23856T4+168327T5+1634317T6+168327pT7+23856p2T8+1872p3T9+210p4T10+9p5T11+p6T12)2 ( 1 + 9 T + 210 T^{2} + 1872 T^{3} + 23856 T^{4} + 168327 T^{5} + 1634317 T^{6} + 168327 p T^{7} + 23856 p^{2} T^{8} + 1872 p^{3} T^{9} + 210 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2}
59 1−24T+81T2+1422T3+5742T4−212163T5−338506T6+10007865T7+84876597T8−387253440T9−9088783722T10+17187276150T11+501529877931T12+17187276150pT13−9088783722p2T14−387253440p3T15+84876597p4T16+10007865p5T17−338506p6T18−212163p7T19+5742p8T20+1422p9T21+81p10T22−24p11T23+p12T24 1 - 24 T + 81 T^{2} + 1422 T^{3} + 5742 T^{4} - 212163 T^{5} - 338506 T^{6} + 10007865 T^{7} + 84876597 T^{8} - 387253440 T^{9} - 9088783722 T^{10} + 17187276150 T^{11} + 501529877931 T^{12} + 17187276150 p T^{13} - 9088783722 p^{2} T^{14} - 387253440 p^{3} T^{15} + 84876597 p^{4} T^{16} + 10007865 p^{5} T^{17} - 338506 p^{6} T^{18} - 212163 p^{7} T^{19} + 5742 p^{8} T^{20} + 1422 p^{9} T^{21} + 81 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24}
61 1−9T−126T2+2587T3−3393T4−203454T5+1847611T6−1670085T7−92881458T8+1260734047T9−5845841559T10−47792391678T11+847544676913T12−47792391678pT13−5845841559p2T14+1260734047p3T15−92881458p4T16−1670085p5T17+1847611p6T18−203454p7T19−3393p8T20+2587p9T21−126p10T22−9p11T23+p12T24 1 - 9 T - 126 T^{2} + 2587 T^{3} - 3393 T^{4} - 203454 T^{5} + 1847611 T^{6} - 1670085 T^{7} - 92881458 T^{8} + 1260734047 T^{9} - 5845841559 T^{10} - 47792391678 T^{11} + 847544676913 T^{12} - 47792391678 p T^{13} - 5845841559 p^{2} T^{14} + 1260734047 p^{3} T^{15} - 92881458 p^{4} T^{16} - 1670085 p^{5} T^{17} + 1847611 p^{6} T^{18} - 203454 p^{7} T^{19} - 3393 p^{8} T^{20} + 2587 p^{9} T^{21} - 126 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24}
67 1−9T−207T2+1570T3+26280T4−135900T5−2376791T6+4866426T7+185070636T8+49256035T9−13259790540T10−6361835544T11+906839351569T12−6361835544pT13−13259790540p2T14+49256035p3T15+185070636p4T16+4866426p5T17−2376791p6T18−135900p7T19+26280p8T20+1570p9T21−207p10T22−9p11T23+p12T24 1 - 9 T - 207 T^{2} + 1570 T^{3} + 26280 T^{4} - 135900 T^{5} - 2376791 T^{6} + 4866426 T^{7} + 185070636 T^{8} + 49256035 T^{9} - 13259790540 T^{10} - 6361835544 T^{11} + 906839351569 T^{12} - 6361835544 p T^{13} - 13259790540 p^{2} T^{14} + 49256035 p^{3} T^{15} + 185070636 p^{4} T^{16} + 4866426 p^{5} T^{17} - 2376791 p^{6} T^{18} - 135900 p^{7} T^{19} + 26280 p^{8} T^{20} + 1570 p^{9} T^{21} - 207 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24}
71 (1+27T+651T2+10071T3+138813T4+1464345T5+13863913T6+1464345pT7+138813p2T8+10071p3T9+651p4T10+27p5T11+p6T12)2 ( 1 + 27 T + 651 T^{2} + 10071 T^{3} + 138813 T^{4} + 1464345 T^{5} + 13863913 T^{6} + 1464345 p T^{7} + 138813 p^{2} T^{8} + 10071 p^{3} T^{9} + 651 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12} )^{2}
73 (1+6T+264T2+1940T3+33111T4+266427T5+2798097T6+266427pT7+33111p2T8+1940p3T9+264p4T10+6p5T11+p6T12)2 ( 1 + 6 T + 264 T^{2} + 1940 T^{3} + 33111 T^{4} + 266427 T^{5} + 2798097 T^{6} + 266427 p T^{7} + 33111 p^{2} T^{8} + 1940 p^{3} T^{9} + 264 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2}
79 1−297T2−140T3+45441T4+43659T5−4620914T6−8783775T7+361258029T8+880271539T9−25327300482T10−33251116437T11+1905100948891T12−33251116437pT13−25327300482p2T14+880271539p3T15+361258029p4T16−8783775p5T17−4620914p6T18+43659p7T19+45441p8T20−140p9T21−297p10T22+p12T24 1 - 297 T^{2} - 140 T^{3} + 45441 T^{4} + 43659 T^{5} - 4620914 T^{6} - 8783775 T^{7} + 361258029 T^{8} + 880271539 T^{9} - 25327300482 T^{10} - 33251116437 T^{11} + 1905100948891 T^{12} - 33251116437 p T^{13} - 25327300482 p^{2} T^{14} + 880271539 p^{3} T^{15} + 361258029 p^{4} T^{16} - 8783775 p^{5} T^{17} - 4620914 p^{6} T^{18} + 43659 p^{7} T^{19} + 45441 p^{8} T^{20} - 140 p^{9} T^{21} - 297 p^{10} T^{22} + p^{12} T^{24}
83 1−12T−189T2+3654T3+6012T4−400611T5+1007894T6+15445989T7−7453143T8+430522020T9−18841320882T10−37064522502T11+2397381696579T12−37064522502pT13−18841320882p2T14+430522020p3T15−7453143p4T16+15445989p5T17+1007894p6T18−400611p7T19+6012p8T20+3654p9T21−189p10T22−12p11T23+p12T24 1 - 12 T - 189 T^{2} + 3654 T^{3} + 6012 T^{4} - 400611 T^{5} + 1007894 T^{6} + 15445989 T^{7} - 7453143 T^{8} + 430522020 T^{9} - 18841320882 T^{10} - 37064522502 T^{11} + 2397381696579 T^{12} - 37064522502 p T^{13} - 18841320882 p^{2} T^{14} + 430522020 p^{3} T^{15} - 7453143 p^{4} T^{16} + 15445989 p^{5} T^{17} + 1007894 p^{6} T^{18} - 400611 p^{7} T^{19} + 6012 p^{8} T^{20} + 3654 p^{9} T^{21} - 189 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24}
89 (1+9T+354T2+2979T3+59703T4+469404T5+6461593T6+469404pT7+59703p2T8+2979p3T9+354p4T10+9p5T11+p6T12)2 ( 1 + 9 T + 354 T^{2} + 2979 T^{3} + 59703 T^{4} + 469404 T^{5} + 6461593 T^{6} + 469404 p T^{7} + 59703 p^{2} T^{8} + 2979 p^{3} T^{9} + 354 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} )^{2}
97 1−378T2+1426T3+75303T4−455031T5−9407879T6+74070450T7+845097381T8−6894378191T9−59284839627T10+283150215594T11+4702665738871T12+283150215594pT13−59284839627p2T14−6894378191p3T15+845097381p4T16+74070450p5T17−9407879p6T18−455031p7T19+75303p8T20+1426p9T21−378p10T22+p12T24 1 - 378 T^{2} + 1426 T^{3} + 75303 T^{4} - 455031 T^{5} - 9407879 T^{6} + 74070450 T^{7} + 845097381 T^{8} - 6894378191 T^{9} - 59284839627 T^{10} + 283150215594 T^{11} + 4702665738871 T^{12} + 283150215594 p T^{13} - 59284839627 p^{2} T^{14} - 6894378191 p^{3} T^{15} + 845097381 p^{4} T^{16} + 74070450 p^{5} T^{17} - 9407879 p^{6} T^{18} - 455031 p^{7} T^{19} + 75303 p^{8} T^{20} + 1426 p^{9} T^{21} - 378 p^{10} T^{22} + p^{12} T^{24}
show more
show less
   L(s)=∏p ∏j=124(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.28088289268776676325307816717, −3.25752156138132276717629517140, −3.12456900298000725644776433212, −3.11281594885382635757352578998, −2.98433430494727650872659411881, −2.85492097412194394475470933253, −2.79064842152111645317478439527, −2.75631751707742437083883440968, −2.58996885623054026515537616075, −2.28348785776799466059598986503, −2.25313220721820892132585390267, −2.24405101095848888143818647200, −2.22192789381039815608951698888, −2.21782955143084303662489989591, −2.07403319719057158188113461016, −1.79643752401356964227417481285, −1.62941111586927446646497615605, −1.52509393782447354299499772960, −1.34277314082369940747271543897, −1.21746995324517052354198969529, −1.05191670232996038524504172276, −0.976260125419474848159466710257, −0.942077806082286805687160254170, −0.826329777473242051696321746943, −0.57693279686071007175702932035, 0.57693279686071007175702932035, 0.826329777473242051696321746943, 0.942077806082286805687160254170, 0.976260125419474848159466710257, 1.05191670232996038524504172276, 1.21746995324517052354198969529, 1.34277314082369940747271543897, 1.52509393782447354299499772960, 1.62941111586927446646497615605, 1.79643752401356964227417481285, 2.07403319719057158188113461016, 2.21782955143084303662489989591, 2.22192789381039815608951698888, 2.24405101095848888143818647200, 2.25313220721820892132585390267, 2.28348785776799466059598986503, 2.58996885623054026515537616075, 2.75631751707742437083883440968, 2.79064842152111645317478439527, 2.85492097412194394475470933253, 2.98433430494727650872659411881, 3.11281594885382635757352578998, 3.12456900298000725644776433212, 3.25752156138132276717629517140, 3.28088289268776676325307816717

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.