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Properties

Label 12-3808e6-1.1-c1e6-0-3
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 − 4·13-s − 16·15-s + 6·17-s + 18·19-s − 24·21-s + 6·23-s − 25-s − 10·27-s + 4·29-s − 8·31-s + 32·33-s + 24·35-s − 8·37-s − 16·39-s − 4·41-s + 20·43-s − 12·45-s + 6·47-s + 21·49-s + 24·51-s + 14·53-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 − 1.10·13-s − 4.13·15-s + 1.45·17-s + 4.12·19-s − 5.23·21-s + 1.25·23-s − 1/5·25-s − 1.92·27-s + 0.742·29-s − 1.43·31-s + 5.57·33-s + 4.05·35-s − 1.31·37-s − 2.56·39-s − 0.624·41-s + 3.04·43-s − 1.78·45-s + 0.875·47-s + 3·49-s + 3.36·51-s + 1.92·53-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 22.6181611722.61816117
L(12)L(\frac12) ≈\approx 22.6181611722.61816117
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+13T2−10pT3+58T4−32pT5+172T6−32p2T7+58p2T8−10p4T9+13p4T10−4p5T11+p6T12 1 - 4 T + 13 T^{2} - 10 p T^{3} + 58 T^{4} - 32 p T^{5} + 172 T^{6} - 32 p^{2} T^{7} + 58 p^{2} T^{8} - 10 p^{4} T^{9} + 13 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
5 1+4T+17T2+38T3+108T4+36pT5+4p3T6+36p2T7+108p2T8+38p3T9+17p4T10+4p5T11+p6T12 1 + 4 T + 17 T^{2} + 38 T^{3} + 108 T^{4} + 36 p T^{5} + 4 p^{3} T^{6} + 36 p^{2} T^{7} + 108 p^{2} T^{8} + 38 p^{3} T^{9} + 17 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
11 1−8T+52T2−20pT3+81pT4−3084T5+10848T6−3084pT7+81p3T8−20p4T9+52p4T10−8p5T11+p6T12 1 - 8 T + 52 T^{2} - 20 p T^{3} + 81 p T^{4} - 3084 T^{5} + 10848 T^{6} - 3084 p T^{7} + 81 p^{3} T^{8} - 20 p^{4} T^{9} + 52 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
13 1+4T+56T2+200T3+1507T4+4660T5+24584T6+4660pT7+1507p2T8+200p3T9+56p4T10+4p5T11+p6T12 1 + 4 T + 56 T^{2} + 200 T^{3} + 1507 T^{4} + 4660 T^{5} + 24584 T^{6} + 4660 p T^{7} + 1507 p^{2} T^{8} + 200 p^{3} T^{9} + 56 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
19 1−18T+202T2−1618T3+10435T4−56040T5+261988T6−56040pT7+10435p2T8−1618p3T9+202p4T10−18p5T11+p6T12 1 - 18 T + 202 T^{2} - 1618 T^{3} + 10435 T^{4} - 56040 T^{5} + 261988 T^{6} - 56040 p T^{7} + 10435 p^{2} T^{8} - 1618 p^{3} T^{9} + 202 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12}
23 1−6T+58T2−270T3+2451T4−8744T5+57028T6−8744pT7+2451p2T8−270p3T9+58p4T10−6p5T11+p6T12 1 - 6 T + 58 T^{2} - 270 T^{3} + 2451 T^{4} - 8744 T^{5} + 57028 T^{6} - 8744 p T^{7} + 2451 p^{2} T^{8} - 270 p^{3} T^{9} + 58 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
29 1−4T+124T2−528T3+7603T4−27764T5+281776T6−27764pT7+7603p2T8−528p3T9+124p4T10−4p5T11+p6T12 1 - 4 T + 124 T^{2} - 528 T^{3} + 7603 T^{4} - 27764 T^{5} + 281776 T^{6} - 27764 p T^{7} + 7603 p^{2} T^{8} - 528 p^{3} T^{9} + 124 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
31 1+8T+135T2+1006T3+8330T4+55450T5+316652T6+55450pT7+8330p2T8+1006p3T9+135p4T10+8p5T11+p6T12 1 + 8 T + 135 T^{2} + 1006 T^{3} + 8330 T^{4} + 55450 T^{5} + 316652 T^{6} + 55450 p T^{7} + 8330 p^{2} T^{8} + 1006 p^{3} T^{9} + 135 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
37 1+8T+118T2+476T3+4883T4+10836T5+160764T6+10836pT7+4883p2T8+476p3T9+118p4T10+8p5T11+p6T12 1 + 8 T + 118 T^{2} + 476 T^{3} + 4883 T^{4} + 10836 T^{5} + 160764 T^{6} + 10836 p T^{7} + 4883 p^{2} T^{8} + 476 p^{3} T^{9} + 118 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
41 1+4T+149T2+286T3+9140T4+1718T5+386364T6+1718pT7+9140p2T8+286p3T9+149p4T10+4p5T11+p6T12 1 + 4 T + 149 T^{2} + 286 T^{3} + 9140 T^{4} + 1718 T^{5} + 386364 T^{6} + 1718 p T^{7} + 9140 p^{2} T^{8} + 286 p^{3} T^{9} + 149 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
43 1−20T+397T2−4688T3+51994T4−419784T5+3162376T6−419784pT7+51994p2T8−4688p3T9+397p4T10−20p5T11+p6T12 1 - 20 T + 397 T^{2} - 4688 T^{3} + 51994 T^{4} - 419784 T^{5} + 3162376 T^{6} - 419784 p T^{7} + 51994 p^{2} T^{8} - 4688 p^{3} T^{9} + 397 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12}
47 1−6T+176T2−866T3+14819T4−63532T5+827864T6−63532pT7+14819p2T8−866p3T9+176p4T10−6p5T11+p6T12 1 - 6 T + 176 T^{2} - 866 T^{3} + 14819 T^{4} - 63532 T^{5} + 827864 T^{6} - 63532 p T^{7} + 14819 p^{2} T^{8} - 866 p^{3} T^{9} + 176 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
53 1−14T+261T2−2956T3+32512T4−277990T5+2263724T6−277990pT7+32512p2T8−2956p3T9+261p4T10−14p5T11+p6T12 1 - 14 T + 261 T^{2} - 2956 T^{3} + 32512 T^{4} - 277990 T^{5} + 2263724 T^{6} - 277990 p T^{7} + 32512 p^{2} T^{8} - 2956 p^{3} T^{9} + 261 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12}
59 1−20T+366T2−4204T3+47463T4−408168T5+3531140T6−408168pT7+47463p2T8−4204p3T9+366p4T10−20p5T11+p6T12 1 - 20 T + 366 T^{2} - 4204 T^{3} + 47463 T^{4} - 408168 T^{5} + 3531140 T^{6} - 408168 p T^{7} + 47463 p^{2} T^{8} - 4204 p^{3} T^{9} + 366 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12}
61 1−2T+269T2−562T3+34032T4−63610T5+2601044T6−63610pT7+34032p2T8−562p3T9+269p4T10−2p5T11+p6T12 1 - 2 T + 269 T^{2} - 562 T^{3} + 34032 T^{4} - 63610 T^{5} + 2601044 T^{6} - 63610 p T^{7} + 34032 p^{2} T^{8} - 562 p^{3} T^{9} + 269 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
67 1−28T+647T2−10040T3+132218T4−1384784T5+12507300T6−1384784pT7+132218p2T8−10040p3T9+647p4T10−28p5T11+p6T12 1 - 28 T + 647 T^{2} - 10040 T^{3} + 132218 T^{4} - 1384784 T^{5} + 12507300 T^{6} - 1384784 p T^{7} + 132218 p^{2} T^{8} - 10040 p^{3} T^{9} + 647 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12}
71 1−18T+342T2−4106T3+50939T4−484328T5+4569916T6−484328pT7+50939p2T8−4106p3T9+342p4T10−18p5T11+p6T12 1 - 18 T + 342 T^{2} - 4106 T^{3} + 50939 T^{4} - 484328 T^{5} + 4569916 T^{6} - 484328 p T^{7} + 50939 p^{2} T^{8} - 4106 p^{3} T^{9} + 342 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12}
73 1−4T+245T2−458T3+26548T4+7374T5+2042124T6+7374pT7+26548p2T8−458p3T9+245p4T10−4p5T11+p6T12 1 - 4 T + 245 T^{2} - 458 T^{3} + 26548 T^{4} + 7374 T^{5} + 2042124 T^{6} + 7374 p T^{7} + 26548 p^{2} T^{8} - 458 p^{3} T^{9} + 245 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
79 1+2T+312T2+646T3+46403T4+83092T5+4426632T6+83092pT7+46403p2T8+646p3T9+312p4T10+2p5T11+p6T12 1 + 2 T + 312 T^{2} + 646 T^{3} + 46403 T^{4} + 83092 T^{5} + 4426632 T^{6} + 83092 p T^{7} + 46403 p^{2} T^{8} + 646 p^{3} T^{9} + 312 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
83 1−20T+416T2−4760T3+61547T4−531412T5+5714376T6−531412pT7+61547p2T8−4760p3T9+416p4T10−20p5T11+p6T12 1 - 20 T + 416 T^{2} - 4760 T^{3} + 61547 T^{4} - 531412 T^{5} + 5714376 T^{6} - 531412 p T^{7} + 61547 p^{2} T^{8} - 4760 p^{3} T^{9} + 416 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12}
89 1+6T+452T2+1886T3+87475T4+263996T5+9819472T6+263996pT7+87475p2T8+1886p3T9+452p4T10+6p5T11+p6T12 1 + 6 T + 452 T^{2} + 1886 T^{3} + 87475 T^{4} + 263996 T^{5} + 9819472 T^{6} + 263996 p T^{7} + 87475 p^{2} T^{8} + 1886 p^{3} T^{9} + 452 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
97 1−30T+669T2−9782T3+130940T4−1420356T5+15225068T6−1420356pT7+130940p2T8−9782p3T9+669p4T10−30p5T11+p6T12 1 - 30 T + 669 T^{2} - 9782 T^{3} + 130940 T^{4} - 1420356 T^{5} + 15225068 T^{6} - 1420356 p T^{7} + 130940 p^{2} T^{8} - 9782 p^{3} T^{9} + 669 p^{4} T^{10} - 30 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

−4.30158779163089325054163801309, −3.85752121360033372576913655774, −3.76105047211732371899603899684, −3.73744379752030841500513750587, −3.73206265168408731468922392765, −3.66583225631843096271255268410, −3.64413995485556317993550868038, −3.51515940593787809576612255703, −3.18808190485466694826530391918, −2.99793209451759325089688961289, −2.95954879626251905735215396341, −2.91882570141122372032465381353, −2.83038416130972536185034211025, −2.49162512720235068673032840135, −2.16026894806286075331545168113, −2.13522910792283483847397317785, −2.12858840602731112202317422397, −2.07944515629037820820180601866, −1.48055897582911244022795010023, −1.23888522283115402199555471604, −1.01716072908136373062109066652, −0.869016709221692137526318871034, −0.68137599494507160814684577803, −0.51291678364496915903414966114, −0.48933382911001243158687814577, 0.48933382911001243158687814577, 0.51291678364496915903414966114, 0.68137599494507160814684577803, 0.869016709221692137526318871034, 1.01716072908136373062109066652, 1.23888522283115402199555471604, 1.48055897582911244022795010023, 2.07944515629037820820180601866, 2.12858840602731112202317422397, 2.13522910792283483847397317785, 2.16026894806286075331545168113, 2.49162512720235068673032840135, 2.83038416130972536185034211025, 2.91882570141122372032465381353, 2.95954879626251905735215396341, 2.99793209451759325089688961289, 3.18808190485466694826530391918, 3.51515940593787809576612255703, 3.64413995485556317993550868038, 3.66583225631843096271255268410, 3.73206265168408731468922392765, 3.73744379752030841500513750587, 3.76105047211732371899603899684, 3.85752121360033372576913655774, 4.30158779163089325054163801309

Graph of the ZZ-function along the critical line

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