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Properties

Label 12-3808e6-1.1-c1e6-0-3
Degree $12$
Conductor $3.049\times 10^{21}$
Sign $1$
Analytic cond. $7.90395\times 10^{8}$
Root an. cond. $5.51425$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30} \cdot 7^{6} \cdot 17^{6}\)
Sign: $1$
Analytic conductor: \(7.90395\times 10^{8}\)
Root analytic conductor: \(5.51425\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(22.61816117\)
\(L(\frac12)\) \(\approx\) \(22.61816117\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( ( 1 + T )^{6} \)
17 \( ( 1 - T )^{6} \)
good3 \( 1 - 4 T + 13 T^{2} - 10 p T^{3} + 58 T^{4} - 32 p T^{5} + 172 T^{6} - 32 p^{2} T^{7} + 58 p^{2} T^{8} - 10 p^{4} T^{9} + 13 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
5 \( 1 + 4 T + 17 T^{2} + 38 T^{3} + 108 T^{4} + 36 p T^{5} + 4 p^{3} T^{6} + 36 p^{2} T^{7} + 108 p^{2} T^{8} + 38 p^{3} T^{9} + 17 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 8 T + 52 T^{2} - 20 p T^{3} + 81 p T^{4} - 3084 T^{5} + 10848 T^{6} - 3084 p T^{7} + 81 p^{3} T^{8} - 20 p^{4} T^{9} + 52 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 4 T + 56 T^{2} + 200 T^{3} + 1507 T^{4} + 4660 T^{5} + 24584 T^{6} + 4660 p T^{7} + 1507 p^{2} T^{8} + 200 p^{3} T^{9} + 56 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 18 T + 202 T^{2} - 1618 T^{3} + 10435 T^{4} - 56040 T^{5} + 261988 T^{6} - 56040 p T^{7} + 10435 p^{2} T^{8} - 1618 p^{3} T^{9} + 202 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 6 T + 58 T^{2} - 270 T^{3} + 2451 T^{4} - 8744 T^{5} + 57028 T^{6} - 8744 p T^{7} + 2451 p^{2} T^{8} - 270 p^{3} T^{9} + 58 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 4 T + 124 T^{2} - 528 T^{3} + 7603 T^{4} - 27764 T^{5} + 281776 T^{6} - 27764 p T^{7} + 7603 p^{2} T^{8} - 528 p^{3} T^{9} + 124 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 8 T + 135 T^{2} + 1006 T^{3} + 8330 T^{4} + 55450 T^{5} + 316652 T^{6} + 55450 p T^{7} + 8330 p^{2} T^{8} + 1006 p^{3} T^{9} + 135 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 8 T + 118 T^{2} + 476 T^{3} + 4883 T^{4} + 10836 T^{5} + 160764 T^{6} + 10836 p T^{7} + 4883 p^{2} T^{8} + 476 p^{3} T^{9} + 118 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 4 T + 149 T^{2} + 286 T^{3} + 9140 T^{4} + 1718 T^{5} + 386364 T^{6} + 1718 p T^{7} + 9140 p^{2} T^{8} + 286 p^{3} T^{9} + 149 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 20 T + 397 T^{2} - 4688 T^{3} + 51994 T^{4} - 419784 T^{5} + 3162376 T^{6} - 419784 p T^{7} + 51994 p^{2} T^{8} - 4688 p^{3} T^{9} + 397 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 6 T + 176 T^{2} - 866 T^{3} + 14819 T^{4} - 63532 T^{5} + 827864 T^{6} - 63532 p T^{7} + 14819 p^{2} T^{8} - 866 p^{3} T^{9} + 176 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 14 T + 261 T^{2} - 2956 T^{3} + 32512 T^{4} - 277990 T^{5} + 2263724 T^{6} - 277990 p T^{7} + 32512 p^{2} T^{8} - 2956 p^{3} T^{9} + 261 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 20 T + 366 T^{2} - 4204 T^{3} + 47463 T^{4} - 408168 T^{5} + 3531140 T^{6} - 408168 p T^{7} + 47463 p^{2} T^{8} - 4204 p^{3} T^{9} + 366 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 2 T + 269 T^{2} - 562 T^{3} + 34032 T^{4} - 63610 T^{5} + 2601044 T^{6} - 63610 p T^{7} + 34032 p^{2} T^{8} - 562 p^{3} T^{9} + 269 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 28 T + 647 T^{2} - 10040 T^{3} + 132218 T^{4} - 1384784 T^{5} + 12507300 T^{6} - 1384784 p T^{7} + 132218 p^{2} T^{8} - 10040 p^{3} T^{9} + 647 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 18 T + 342 T^{2} - 4106 T^{3} + 50939 T^{4} - 484328 T^{5} + 4569916 T^{6} - 484328 p T^{7} + 50939 p^{2} T^{8} - 4106 p^{3} T^{9} + 342 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 4 T + 245 T^{2} - 458 T^{3} + 26548 T^{4} + 7374 T^{5} + 2042124 T^{6} + 7374 p T^{7} + 26548 p^{2} T^{8} - 458 p^{3} T^{9} + 245 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 2 T + 312 T^{2} + 646 T^{3} + 46403 T^{4} + 83092 T^{5} + 4426632 T^{6} + 83092 p T^{7} + 46403 p^{2} T^{8} + 646 p^{3} T^{9} + 312 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 20 T + 416 T^{2} - 4760 T^{3} + 61547 T^{4} - 531412 T^{5} + 5714376 T^{6} - 531412 p T^{7} + 61547 p^{2} T^{8} - 4760 p^{3} T^{9} + 416 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 6 T + 452 T^{2} + 1886 T^{3} + 87475 T^{4} + 263996 T^{5} + 9819472 T^{6} + 263996 p T^{7} + 87475 p^{2} T^{8} + 1886 p^{3} T^{9} + 452 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 30 T + 669 T^{2} - 9782 T^{3} + 130940 T^{4} - 1420356 T^{5} + 15225068 T^{6} - 1420356 p T^{7} + 130940 p^{2} T^{8} - 9782 p^{3} T^{9} + 669 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line

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