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Properties

Label 18-2523e9-1.1-c1e9-0-0
Degree 1818
Conductor 4.142×10304.142\times 10^{30}
Sign 11
Analytic cond. 5.46700×10115.46700\times 10^{11}
Root an. cond. 4.488454.48845
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 9·3-s + 9·4-s − 4·5-s + 45·6-s + 5·7-s + 8·8-s + 45·9-s − 20·10-s − 11-s + 81·12-s + 13-s + 25·14-s − 36·15-s + 15·16-s + 2·17-s + 225·18-s + 9·19-s − 36·20-s + 45·21-s − 5·22-s − 4·23-s + 72·24-s − 14·25-s + 5·26-s + 165·27-s + 45·28-s + ⋯
L(s)  = 1  + 3.53·2-s + 5.19·3-s + 9/2·4-s − 1.78·5-s + 18.3·6-s + 1.88·7-s + 2.82·8-s + 15·9-s − 6.32·10-s − 0.301·11-s + 23.3·12-s + 0.277·13-s + 6.68·14-s − 9.29·15-s + 15/4·16-s + 0.485·17-s + 53.0·18-s + 2.06·19-s − 8.04·20-s + 9.81·21-s − 1.06·22-s − 0.834·23-s + 14.6·24-s − 2.79·25-s + 0.980·26-s + 31.7·27-s + 8.50·28-s + ⋯

Functional equation

Λ(s)=((39⋅2918)s/2 ΓC(s)9 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((39⋅2918)s/2 ΓC(s+1/2)9 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 29^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1818
Conductor: 39⋅29183^{9} \cdot 29^{18}
Sign: 11
Analytic conductor: 5.46700×10115.46700\times 10^{11}
Root analytic conductor: 4.488454.48845
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (18, 39⋅2918, ( :[1/2]9), 1)(18,\ 3^{9} \cdot 29^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 627.5357647627.5357647
L(12)L(\frac12) ≈\approx 627.5357647627.5357647
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−T)9 ( 1 - T )^{9}
29 1 1
good2 1−5T+p4T2−43T3+3p5T4−187T5+335T6−137p2T7+105p3T8−1225T9+105p4T10−137p4T11+335p3T12−187p4T13+3p10T14−43p6T15+p11T16−5p8T17+p9T18 1 - 5 T + p^{4} T^{2} - 43 T^{3} + 3 p^{5} T^{4} - 187 T^{5} + 335 T^{6} - 137 p^{2} T^{7} + 105 p^{3} T^{8} - 1225 T^{9} + 105 p^{4} T^{10} - 137 p^{4} T^{11} + 335 p^{3} T^{12} - 187 p^{4} T^{13} + 3 p^{10} T^{14} - 43 p^{6} T^{15} + p^{11} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18}
5 1+4T+6pT2+92T3+433T4+1113T5+4008T6+8849T7+26633T8+51301T9+26633pT10+8849p2T11+4008p3T12+1113p4T13+433p5T14+92p6T15+6p8T16+4p8T17+p9T18 1 + 4 T + 6 p T^{2} + 92 T^{3} + 433 T^{4} + 1113 T^{5} + 4008 T^{6} + 8849 T^{7} + 26633 T^{8} + 51301 T^{9} + 26633 p T^{10} + 8849 p^{2} T^{11} + 4008 p^{3} T^{12} + 1113 p^{4} T^{13} + 433 p^{5} T^{14} + 92 p^{6} T^{15} + 6 p^{8} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18}
7 1−5T+47T2−172T3+137pT4−2893T5+12304T6−4576pT7+113343T8−258348T9+113343pT10−4576p3T11+12304p3T12−2893p4T13+137p6T14−172p6T15+47p7T16−5p8T17+p9T18 1 - 5 T + 47 T^{2} - 172 T^{3} + 137 p T^{4} - 2893 T^{5} + 12304 T^{6} - 4576 p T^{7} + 113343 T^{8} - 258348 T^{9} + 113343 p T^{10} - 4576 p^{3} T^{11} + 12304 p^{3} T^{12} - 2893 p^{4} T^{13} + 137 p^{6} T^{14} - 172 p^{6} T^{15} + 47 p^{7} T^{16} - 5 p^{8} T^{17} + p^{9} T^{18}
11 1+T+49T2+1099T4−1345T5+15750T6−39934T7+180715T8−581228T9+180715pT10−39934p2T11+15750p3T12−1345p4T13+1099p5T14+49p7T16+p8T17+p9T18 1 + T + 49 T^{2} + 1099 T^{4} - 1345 T^{5} + 15750 T^{6} - 39934 T^{7} + 180715 T^{8} - 581228 T^{9} + 180715 p T^{10} - 39934 p^{2} T^{11} + 15750 p^{3} T^{12} - 1345 p^{4} T^{13} + 1099 p^{5} T^{14} + 49 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18}
13 1−T+30T2−149T3+716T4−2996T5+20065T6−53796T7+270263T8−1030329T9+270263pT10−53796p2T11+20065p3T12−2996p4T13+716p5T14−149p6T15+30p7T16−p8T17+p9T18 1 - T + 30 T^{2} - 149 T^{3} + 716 T^{4} - 2996 T^{5} + 20065 T^{6} - 53796 T^{7} + 270263 T^{8} - 1030329 T^{9} + 270263 p T^{10} - 53796 p^{2} T^{11} + 20065 p^{3} T^{12} - 2996 p^{4} T^{13} + 716 p^{5} T^{14} - 149 p^{6} T^{15} + 30 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18}
17 1−2T+90T2−154T3+4290T4−6733T5+135098T6−189810T7+3077115T8−3827129T9+3077115pT10−189810p2T11+135098p3T12−6733p4T13+4290p5T14−154p6T15+90p7T16−2p8T17+p9T18 1 - 2 T + 90 T^{2} - 154 T^{3} + 4290 T^{4} - 6733 T^{5} + 135098 T^{6} - 189810 T^{7} + 3077115 T^{8} - 3827129 T^{9} + 3077115 p T^{10} - 189810 p^{2} T^{11} + 135098 p^{3} T^{12} - 6733 p^{4} T^{13} + 4290 p^{5} T^{14} - 154 p^{6} T^{15} + 90 p^{7} T^{16} - 2 p^{8} T^{17} + p^{9} T^{18}
19 1−9T+92T2−594T3+4533T4−24609T5+146900T6−687176T7+3585054T8−14915472T9+3585054pT10−687176p2T11+146900p3T12−24609p4T13+4533p5T14−594p6T15+92p7T16−9p8T17+p9T18 1 - 9 T + 92 T^{2} - 594 T^{3} + 4533 T^{4} - 24609 T^{5} + 146900 T^{6} - 687176 T^{7} + 3585054 T^{8} - 14915472 T^{9} + 3585054 p T^{10} - 687176 p^{2} T^{11} + 146900 p^{3} T^{12} - 24609 p^{4} T^{13} + 4533 p^{5} T^{14} - 594 p^{6} T^{15} + 92 p^{7} T^{16} - 9 p^{8} T^{17} + p^{9} T^{18}
23 1+4T+88T2+262T3+3461T4+8620T5+92680T6+254144T7+2259442T8+6737116T9+2259442pT10+254144p2T11+92680p3T12+8620p4T13+3461p5T14+262p6T15+88p7T16+4p8T17+p9T18 1 + 4 T + 88 T^{2} + 262 T^{3} + 3461 T^{4} + 8620 T^{5} + 92680 T^{6} + 254144 T^{7} + 2259442 T^{8} + 6737116 T^{9} + 2259442 p T^{10} + 254144 p^{2} T^{11} + 92680 p^{3} T^{12} + 8620 p^{4} T^{13} + 3461 p^{5} T^{14} + 262 p^{6} T^{15} + 88 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18}
31 1−8T+223T2−1452T3+22492T4−124237T5+1395045T6−6653364T7+59826447T8−245448446T9+59826447pT10−6653364p2T11+1395045p3T12−124237p4T13+22492p5T14−1452p6T15+223p7T16−8p8T17+p9T18 1 - 8 T + 223 T^{2} - 1452 T^{3} + 22492 T^{4} - 124237 T^{5} + 1395045 T^{6} - 6653364 T^{7} + 59826447 T^{8} - 245448446 T^{9} + 59826447 p T^{10} - 6653364 p^{2} T^{11} + 1395045 p^{3} T^{12} - 124237 p^{4} T^{13} + 22492 p^{5} T^{14} - 1452 p^{6} T^{15} + 223 p^{7} T^{16} - 8 p^{8} T^{17} + p^{9} T^{18}
37 1−27T+544T2−7852T3+96462T4−991317T5+9025832T6−71977894T7+517158242T8−3306525525T9+517158242pT10−71977894p2T11+9025832p3T12−991317p4T13+96462p5T14−7852p6T15+544p7T16−27p8T17+p9T18 1 - 27 T + 544 T^{2} - 7852 T^{3} + 96462 T^{4} - 991317 T^{5} + 9025832 T^{6} - 71977894 T^{7} + 517158242 T^{8} - 3306525525 T^{9} + 517158242 p T^{10} - 71977894 p^{2} T^{11} + 9025832 p^{3} T^{12} - 991317 p^{4} T^{13} + 96462 p^{5} T^{14} - 7852 p^{6} T^{15} + 544 p^{7} T^{16} - 27 p^{8} T^{17} + p^{9} T^{18}
41 1−12T+226T2−2136T3+24552T4−199425T5+1767092T6−12614876T7+94224175T8−594125521T9+94224175pT10−12614876p2T11+1767092p3T12−199425p4T13+24552p5T14−2136p6T15+226p7T16−12p8T17+p9T18 1 - 12 T + 226 T^{2} - 2136 T^{3} + 24552 T^{4} - 199425 T^{5} + 1767092 T^{6} - 12614876 T^{7} + 94224175 T^{8} - 594125521 T^{9} + 94224175 p T^{10} - 12614876 p^{2} T^{11} + 1767092 p^{3} T^{12} - 199425 p^{4} T^{13} + 24552 p^{5} T^{14} - 2136 p^{6} T^{15} + 226 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18}
43 1−16T+349T2−4222T3+53464T4−515381T5+4875605T6−38702340T7+298086737T8−1984494250T9+298086737pT10−38702340p2T11+4875605p3T12−515381p4T13+53464p5T14−4222p6T15+349p7T16−16p8T17+p9T18 1 - 16 T + 349 T^{2} - 4222 T^{3} + 53464 T^{4} - 515381 T^{5} + 4875605 T^{6} - 38702340 T^{7} + 298086737 T^{8} - 1984494250 T^{9} + 298086737 p T^{10} - 38702340 p^{2} T^{11} + 4875605 p^{3} T^{12} - 515381 p^{4} T^{13} + 53464 p^{5} T^{14} - 4222 p^{6} T^{15} + 349 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18}
47 1+8T+380T2+2565T3+65031T4+372743T5+6647480T6+32363927T7+451278644T8+1849968562T9+451278644pT10+32363927p2T11+6647480p3T12+372743p4T13+65031p5T14+2565p6T15+380p7T16+8p8T17+p9T18 1 + 8 T + 380 T^{2} + 2565 T^{3} + 65031 T^{4} + 372743 T^{5} + 6647480 T^{6} + 32363927 T^{7} + 451278644 T^{8} + 1849968562 T^{9} + 451278644 p T^{10} + 32363927 p^{2} T^{11} + 6647480 p^{3} T^{12} + 372743 p^{4} T^{13} + 65031 p^{5} T^{14} + 2565 p^{6} T^{15} + 380 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18}
53 1+8T+273T2+2044T3+36211T4+260714T5+3153661T6+21870087T7+207103767T8+1338205431T9+207103767pT10+21870087p2T11+3153661p3T12+260714p4T13+36211p5T14+2044p6T15+273p7T16+8p8T17+p9T18 1 + 8 T + 273 T^{2} + 2044 T^{3} + 36211 T^{4} + 260714 T^{5} + 3153661 T^{6} + 21870087 T^{7} + 207103767 T^{8} + 1338205431 T^{9} + 207103767 p T^{10} + 21870087 p^{2} T^{11} + 3153661 p^{3} T^{12} + 260714 p^{4} T^{13} + 36211 p^{5} T^{14} + 2044 p^{6} T^{15} + 273 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18}
59 1+16T+393T2+3630T3+49303T4+216050T5+1990154T6−10149150T7−52334721T8−1597546588T9−52334721pT10−10149150p2T11+1990154p3T12+216050p4T13+49303p5T14+3630p6T15+393p7T16+16p8T17+p9T18 1 + 16 T + 393 T^{2} + 3630 T^{3} + 49303 T^{4} + 216050 T^{5} + 1990154 T^{6} - 10149150 T^{7} - 52334721 T^{8} - 1597546588 T^{9} - 52334721 p T^{10} - 10149150 p^{2} T^{11} + 1990154 p^{3} T^{12} + 216050 p^{4} T^{13} + 49303 p^{5} T^{14} + 3630 p^{6} T^{15} + 393 p^{7} T^{16} + 16 p^{8} T^{17} + p^{9} T^{18}
61 1−21T+593T2−9039T3+147416T4−1763827T5+20995514T6−203924756T7+1916422141T8−15273534255T9+1916422141pT10−203924756p2T11+20995514p3T12−1763827p4T13+147416p5T14−9039p6T15+593p7T16−21p8T17+p9T18 1 - 21 T + 593 T^{2} - 9039 T^{3} + 147416 T^{4} - 1763827 T^{5} + 20995514 T^{6} - 203924756 T^{7} + 1916422141 T^{8} - 15273534255 T^{9} + 1916422141 p T^{10} - 203924756 p^{2} T^{11} + 20995514 p^{3} T^{12} - 1763827 p^{4} T^{13} + 147416 p^{5} T^{14} - 9039 p^{6} T^{15} + 593 p^{7} T^{16} - 21 p^{8} T^{17} + p^{9} T^{18}
67 1−3T+379T2−1384T3+72420T4−274006T5+9106607T6−32870880T7+821083453T8−2656323014T9+821083453pT10−32870880p2T11+9106607p3T12−274006p4T13+72420p5T14−1384p6T15+379p7T16−3p8T17+p9T18 1 - 3 T + 379 T^{2} - 1384 T^{3} + 72420 T^{4} - 274006 T^{5} + 9106607 T^{6} - 32870880 T^{7} + 821083453 T^{8} - 2656323014 T^{9} + 821083453 p T^{10} - 32870880 p^{2} T^{11} + 9106607 p^{3} T^{12} - 274006 p^{4} T^{13} + 72420 p^{5} T^{14} - 1384 p^{6} T^{15} + 379 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18}
71 1+33T+817T2+13790T3+199530T4+2418510T5+27125871T6+276226520T7+2646386681T8+23124430206T9+2646386681pT10+276226520p2T11+27125871p3T12+2418510p4T13+199530p5T14+13790p6T15+817p7T16+33p8T17+p9T18 1 + 33 T + 817 T^{2} + 13790 T^{3} + 199530 T^{4} + 2418510 T^{5} + 27125871 T^{6} + 276226520 T^{7} + 2646386681 T^{8} + 23124430206 T^{9} + 2646386681 p T^{10} + 276226520 p^{2} T^{11} + 27125871 p^{3} T^{12} + 2418510 p^{4} T^{13} + 199530 p^{5} T^{14} + 13790 p^{6} T^{15} + 817 p^{7} T^{16} + 33 p^{8} T^{17} + p^{9} T^{18}
73 1−3T+369T2−774T3+68199T4−91024T5+8591261T6−8183155T7+814407150T8−656168425T9+814407150pT10−8183155p2T11+8591261p3T12−91024p4T13+68199p5T14−774p6T15+369p7T16−3p8T17+p9T18 1 - 3 T + 369 T^{2} - 774 T^{3} + 68199 T^{4} - 91024 T^{5} + 8591261 T^{6} - 8183155 T^{7} + 814407150 T^{8} - 656168425 T^{9} + 814407150 p T^{10} - 8183155 p^{2} T^{11} + 8591261 p^{3} T^{12} - 91024 p^{4} T^{13} + 68199 p^{5} T^{14} - 774 p^{6} T^{15} + 369 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18}
79 1−3T+439T2−408T3+88556T4+87082T5+11391461T6+27467106T7+1100412555T8+3084319398T9+1100412555pT10+27467106p2T11+11391461p3T12+87082p4T13+88556p5T14−408p6T15+439p7T16−3p8T17+p9T18 1 - 3 T + 439 T^{2} - 408 T^{3} + 88556 T^{4} + 87082 T^{5} + 11391461 T^{6} + 27467106 T^{7} + 1100412555 T^{8} + 3084319398 T^{9} + 1100412555 p T^{10} + 27467106 p^{2} T^{11} + 11391461 p^{3} T^{12} + 87082 p^{4} T^{13} + 88556 p^{5} T^{14} - 408 p^{6} T^{15} + 439 p^{7} T^{16} - 3 p^{8} T^{17} + p^{9} T^{18}
83 1−13T+543T2−7006T3+147206T4−1731416T5+25372343T6−259949004T7+2994947111T8−26073244458T9+2994947111pT10−259949004p2T11+25372343p3T12−1731416p4T13+147206p5T14−7006p6T15+543p7T16−13p8T17+p9T18 1 - 13 T + 543 T^{2} - 7006 T^{3} + 147206 T^{4} - 1731416 T^{5} + 25372343 T^{6} - 259949004 T^{7} + 2994947111 T^{8} - 26073244458 T^{9} + 2994947111 p T^{10} - 259949004 p^{2} T^{11} + 25372343 p^{3} T^{12} - 1731416 p^{4} T^{13} + 147206 p^{5} T^{14} - 7006 p^{6} T^{15} + 543 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18}
89 1−6T+228T2−913T3+24716T4+30276T5+2218155T6+9025240T7+212571166T8+639205597T9+212571166pT10+9025240p2T11+2218155p3T12+30276p4T13+24716p5T14−913p6T15+228p7T16−6p8T17+p9T18 1 - 6 T + 228 T^{2} - 913 T^{3} + 24716 T^{4} + 30276 T^{5} + 2218155 T^{6} + 9025240 T^{7} + 212571166 T^{8} + 639205597 T^{9} + 212571166 p T^{10} + 9025240 p^{2} T^{11} + 2218155 p^{3} T^{12} + 30276 p^{4} T^{13} + 24716 p^{5} T^{14} - 913 p^{6} T^{15} + 228 p^{7} T^{16} - 6 p^{8} T^{17} + p^{9} T^{18}
97 1−4T+518T2−2545T3+130982T4−678282T5+21974185T6−107665712T7+2747966160T8−12054120693T9+2747966160pT10−107665712p2T11+21974185p3T12−678282p4T13+130982p5T14−2545p6T15+518p7T16−4p8T17+p9T18 1 - 4 T + 518 T^{2} - 2545 T^{3} + 130982 T^{4} - 678282 T^{5} + 21974185 T^{6} - 107665712 T^{7} + 2747966160 T^{8} - 12054120693 T^{9} + 2747966160 p T^{10} - 107665712 p^{2} T^{11} + 21974185 p^{3} T^{12} - 678282 p^{4} T^{13} + 130982 p^{5} T^{14} - 2545 p^{6} T^{15} + 518 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18}
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   L(s)=∏p ∏j=118(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.43448914141439926461674312816, −3.40230527357239877693083830086, −3.24817154899435048336131539653, −3.14682224521119589944545929300, −3.14610368748903989360520842296, −3.02201255042069963681506184636, −2.90097920997988230652501350914, −2.68188503314939395436645158944, −2.56615714909296442090323605982, −2.50901362565075512605969087619, −2.48011741041165369734746377531, −2.24123475052888654306924849141, −2.15217765472860767494570991290, −2.05643331633323160761695515692, −1.83479030865817921238710486721, −1.79528675127940766775274433485, −1.56704000455200897424971009654, −1.54636845532489597001089895428, −1.49190814597990825638163341905, −1.19977192444543495548080064216, −1.11711739091424323293047672640, −0.951545780565455695204273454301, −0.818623466191123082196960760735, −0.58088187010590330080946710141, −0.20132559531369548418947833989, 0.20132559531369548418947833989, 0.58088187010590330080946710141, 0.818623466191123082196960760735, 0.951545780565455695204273454301, 1.11711739091424323293047672640, 1.19977192444543495548080064216, 1.49190814597990825638163341905, 1.54636845532489597001089895428, 1.56704000455200897424971009654, 1.79528675127940766775274433485, 1.83479030865817921238710486721, 2.05643331633323160761695515692, 2.15217765472860767494570991290, 2.24123475052888654306924849141, 2.48011741041165369734746377531, 2.50901362565075512605969087619, 2.56615714909296442090323605982, 2.68188503314939395436645158944, 2.90097920997988230652501350914, 3.02201255042069963681506184636, 3.14610368748903989360520842296, 3.14682224521119589944545929300, 3.24817154899435048336131539653, 3.40230527357239877693083830086, 3.43448914141439926461674312816

Graph of the ZZ-function along the critical line

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