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Properties

Label 12-3808e6-1.1-c1e6-0-4
Degree 1212
Conductor 3.049×10213.049\times 10^{21}
Sign 11
Analytic cond. 7.90395×1087.90395\times 10^{8}
Root an. cond. 5.514255.51425
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 4·5-s + 6·7-s + 3·9-s + 8·11-s + 8·13-s + 16·15-s − 6·17-s + 6·19-s + 24·21-s + 18·23-s − 25-s − 6·27-s − 8·29-s − 4·31-s + 32·33-s + 24·35-s − 8·37-s + 32·39-s + 16·41-s + 12·43-s + 12·45-s + 18·47-s + 21·49-s − 24·51-s − 2·53-s + 32·55-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.78·5-s + 2.26·7-s + 9-s + 2.41·11-s + 2.21·13-s + 4.13·15-s − 1.45·17-s + 1.37·19-s + 5.23·21-s + 3.75·23-s − 1/5·25-s − 1.15·27-s − 1.48·29-s − 0.718·31-s + 5.57·33-s + 4.05·35-s − 1.31·37-s + 5.12·39-s + 2.49·41-s + 1.82·43-s + 1.78·45-s + 2.62·47-s + 3·49-s − 3.36·51-s − 0.274·53-s + 4.31·55-s + ⋯

Functional equation

Λ(s)=((230⋅76⋅176)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((230⋅76⋅176)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 230⋅76⋅1762^{30} \cdot 7^{6} \cdot 17^{6}
Sign: 11
Analytic conductor: 7.90395×1087.90395\times 10^{8}
Root analytic conductor: 5.514255.51425
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 230⋅76⋅176, ( :[1/2]6), 1)(12,\ 2^{30} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 192.8957297192.8957297
L(12)L(\frac12) ≈\approx 192.8957297192.8957297
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)
bad2 1 1
7 (1−T)6 ( 1 - T )^{6}
17 (1+T)6 ( 1 + T )^{6}
good3 1−4T+13T2−34T3+26pT4−164T5+304T6−164pT7+26p3T8−34p3T9+13p4T10−4p5T11+p6T12 1 - 4 T + 13 T^{2} - 34 T^{3} + 26 p T^{4} - 164 T^{5} + 304 T^{6} - 164 p T^{7} + 26 p^{3} T^{8} - 34 p^{3} T^{9} + 13 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
5 1−4T+17T2−38T3+24pT4−216T5+616T6−216pT7+24p3T8−38p3T9+17p4T10−4p5T11+p6T12 1 - 4 T + 17 T^{2} - 38 T^{3} + 24 p T^{4} - 216 T^{5} + 616 T^{6} - 216 p T^{7} + 24 p^{3} T^{8} - 38 p^{3} T^{9} + 17 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
11 1−8T+64T2−316T3+1571T4−5772T5+21704T6−5772pT7+1571p2T8−316p3T9+64p4T10−8p5T11+p6T12 1 - 8 T + 64 T^{2} - 316 T^{3} + 1571 T^{4} - 5772 T^{5} + 21704 T^{6} - 5772 p T^{7} + 1571 p^{2} T^{8} - 316 p^{3} T^{9} + 64 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
13 1−8T+56T2−252T3+1139T4−3820T5+14824T6−3820pT7+1139p2T8−252p3T9+56p4T10−8p5T11+p6T12 1 - 8 T + 56 T^{2} - 252 T^{3} + 1139 T^{4} - 3820 T^{5} + 14824 T^{6} - 3820 p T^{7} + 1139 p^{2} T^{8} - 252 p^{3} T^{9} + 56 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
19 1−6T+106T2−510T3+4803T4−18304T5+119684T6−18304pT7+4803p2T8−510p3T9+106p4T10−6p5T11+p6T12 1 - 6 T + 106 T^{2} - 510 T^{3} + 4803 T^{4} - 18304 T^{5} + 119684 T^{6} - 18304 p T^{7} + 4803 p^{2} T^{8} - 510 p^{3} T^{9} + 106 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
23 1−18T+250T2−102pT3+18403T4−114008T5+606116T6−114008pT7+18403p2T8−102p4T9+250p4T10−18p5T11+p6T12 1 - 18 T + 250 T^{2} - 102 p T^{3} + 18403 T^{4} - 114008 T^{5} + 606116 T^{6} - 114008 p T^{7} + 18403 p^{2} T^{8} - 102 p^{4} T^{9} + 250 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12}
29 1+8T+72T2+500T3+4203T4+21724T5+130584T6+21724pT7+4203p2T8+500p3T9+72p4T10+8p5T11+p6T12 1 + 8 T + 72 T^{2} + 500 T^{3} + 4203 T^{4} + 21724 T^{5} + 130584 T^{6} + 21724 p T^{7} + 4203 p^{2} T^{8} + 500 p^{3} T^{9} + 72 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
31 1+4T+91T2+386T3+4658T4+19242T5+161892T6+19242pT7+4658p2T8+386p3T9+91p4T10+4p5T11+p6T12 1 + 4 T + 91 T^{2} + 386 T^{3} + 4658 T^{4} + 19242 T^{5} + 161892 T^{6} + 19242 p T^{7} + 4658 p^{2} T^{8} + 386 p^{3} T^{9} + 91 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
37 1+8T+170T2+1148T3+13347T4+74676T5+624612T6+74676pT7+13347p2T8+1148p3T9+170p4T10+8p5T11+p6T12 1 + 8 T + 170 T^{2} + 1148 T^{3} + 13347 T^{4} + 74676 T^{5} + 624612 T^{6} + 74676 p T^{7} + 13347 p^{2} T^{8} + 1148 p^{3} T^{9} + 170 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
41 1−16T+193T2−1934T3+18548T4−136730T5+924660T6−136730pT7+18548p2T8−1934p3T9+193p4T10−16p5T11+p6T12 1 - 16 T + 193 T^{2} - 1934 T^{3} + 18548 T^{4} - 136730 T^{5} + 924660 T^{6} - 136730 p T^{7} + 18548 p^{2} T^{8} - 1934 p^{3} T^{9} + 193 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12}
43 1−12T+141T2−32pT3+10410T4−71640T5+524920T6−71640pT7+10410p2T8−32p4T9+141p4T10−12p5T11+p6T12 1 - 12 T + 141 T^{2} - 32 p T^{3} + 10410 T^{4} - 71640 T^{5} + 524920 T^{6} - 71640 p T^{7} + 10410 p^{2} T^{8} - 32 p^{4} T^{9} + 141 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}
47 1−18T+320T2−3582T3+37667T4−6596pT5+2354680T6−6596p2T7+37667p2T8−3582p3T9+320p4T10−18p5T11+p6T12 1 - 18 T + 320 T^{2} - 3582 T^{3} + 37667 T^{4} - 6596 p T^{5} + 2354680 T^{6} - 6596 p^{2} T^{7} + 37667 p^{2} T^{8} - 3582 p^{3} T^{9} + 320 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12}
53 1+2T+157T2+52T3+14528T4+2178T5+17548pT6+2178pT7+14528p2T8+52p3T9+157p4T10+2p5T11+p6T12 1 + 2 T + 157 T^{2} + 52 T^{3} + 14528 T^{4} + 2178 T^{5} + 17548 p T^{6} + 2178 p T^{7} + 14528 p^{2} T^{8} + 52 p^{3} T^{9} + 157 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
59 1−16T+358T2−4048T3+52039T4−438144T5+4059220T6−438144pT7+52039p2T8−4048p3T9+358p4T10−16p5T11+p6T12 1 - 16 T + 358 T^{2} - 4048 T^{3} + 52039 T^{4} - 438144 T^{5} + 4059220 T^{6} - 438144 p T^{7} + 52039 p^{2} T^{8} - 4048 p^{3} T^{9} + 358 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12}
61 1−6T+213T2−750T3+21092T4−53490T5+1483288T6−53490pT7+21092p2T8−750p3T9+213p4T10−6p5T11+p6T12 1 - 6 T + 213 T^{2} - 750 T^{3} + 21092 T^{4} - 53490 T^{5} + 1483288 T^{6} - 53490 p T^{7} + 21092 p^{2} T^{8} - 750 p^{3} T^{9} + 213 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
67 1−12T+247T2−2840T3+33674T4−304768T5+2894388T6−304768pT7+33674p2T8−2840p3T9+247p4T10−12p5T11+p6T12 1 - 12 T + 247 T^{2} - 2840 T^{3} + 33674 T^{4} - 304768 T^{5} + 2894388 T^{6} - 304768 p T^{7} + 33674 p^{2} T^{8} - 2840 p^{3} T^{9} + 247 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}
71 1+2T+150T2−614T3+12491T4−73880T5+1138268T6−73880pT7+12491p2T8−614p3T9+150p4T10+2p5T11+p6T12 1 + 2 T + 150 T^{2} - 614 T^{3} + 12491 T^{4} - 73880 T^{5} + 1138268 T^{6} - 73880 p T^{7} + 12491 p^{2} T^{8} - 614 p^{3} T^{9} + 150 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
73 1−8T+161T2−1310T3+11180T4−115922T5+573412T6−115922pT7+11180p2T8−1310p3T9+161p4T10−8p5T11+p6T12 1 - 8 T + 161 T^{2} - 1310 T^{3} + 11180 T^{4} - 115922 T^{5} + 573412 T^{6} - 115922 p T^{7} + 11180 p^{2} T^{8} - 1310 p^{3} T^{9} + 161 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
79 1+18T+288T2+2302T3+24499T4+167772T5+2003480T6+167772pT7+24499p2T8+2302p3T9+288p4T10+18p5T11+p6T12 1 + 18 T + 288 T^{2} + 2302 T^{3} + 24499 T^{4} + 167772 T^{5} + 2003480 T^{6} + 167772 p T^{7} + 24499 p^{2} T^{8} + 2302 p^{3} T^{9} + 288 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12}
83 1−28T+624T2−10128T3+136411T4−1560116T5+15237096T6−1560116pT7+136411p2T8−10128p3T9+624p4T10−28p5T11+p6T12 1 - 28 T + 624 T^{2} - 10128 T^{3} + 136411 T^{4} - 1560116 T^{5} + 15237096 T^{6} - 1560116 p T^{7} + 136411 p^{2} T^{8} - 10128 p^{3} T^{9} + 624 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12}
89 1+2T+380T2+530T3+70659T4+76140T5+7900736T6+76140pT7+70659p2T8+530p3T9+380p4T10+2p5T11+p6T12 1 + 2 T + 380 T^{2} + 530 T^{3} + 70659 T^{4} + 76140 T^{5} + 7900736 T^{6} + 76140 p T^{7} + 70659 p^{2} T^{8} + 530 p^{3} T^{9} + 380 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
97 1+18T+537T2+7646T3+124332T4+1388888T5+15835972T6+1388888pT7+124332p2T8+7646p3T9+537p4T10+18p5T11+p6T12 1 + 18 T + 537 T^{2} + 7646 T^{3} + 124332 T^{4} + 1388888 T^{5} + 15835972 T^{6} + 1388888 p T^{7} + 124332 p^{2} T^{8} + 7646 p^{3} T^{9} + 537 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.11625244001601457786529108078, −4.10207269389219638472301980362, −4.06052625169988615320999152533, −3.88929291074873666133970829246, −3.87709771385512916219780849497, −3.65631802320396844894244554111, −3.57616133006817088611326683391, −3.27311350614320822700294330847, −3.16501878222752860108265975915, −2.98211709314446437782102382137, −2.82725595342919054570743658126, −2.75943651728509294832177774746, −2.68892715145109033232233148226, −2.34326434697725104640803437394, −2.08025920097999169492943297324, −2.07699634726073209072498801487, −1.97286296963843621883810294137, −1.86356518299678695951201277505, −1.71686010631985819566886202780, −1.46194624766015277001237384911, −1.16598987010097829758141949310, −0.978888938983805615210256949315, −0.850227496460415026977711207811, −0.73022592794415503579622005126, −0.66353426530292426963187070118, 0.66353426530292426963187070118, 0.73022592794415503579622005126, 0.850227496460415026977711207811, 0.978888938983805615210256949315, 1.16598987010097829758141949310, 1.46194624766015277001237384911, 1.71686010631985819566886202780, 1.86356518299678695951201277505, 1.97286296963843621883810294137, 2.07699634726073209072498801487, 2.08025920097999169492943297324, 2.34326434697725104640803437394, 2.68892715145109033232233148226, 2.75943651728509294832177774746, 2.82725595342919054570743658126, 2.98211709314446437782102382137, 3.16501878222752860108265975915, 3.27311350614320822700294330847, 3.57616133006817088611326683391, 3.65631802320396844894244554111, 3.87709771385512916219780849497, 3.88929291074873666133970829246, 4.06052625169988615320999152533, 4.10207269389219638472301980362, 4.11625244001601457786529108078

Graph of the ZZ-function along the critical line

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