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Properties

Label 12-6018e6-1.1-c1e6-0-1
Degree $12$
Conductor $4.750\times 10^{22}$
Sign $1$
Analytic cond. $1.23133\times 10^{10}$
Root an. cond. $6.93209$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 6·3-s + 21·4-s − 3·5-s + 36·6-s − 7·7-s + 56·8-s + 21·9-s − 18·10-s − 8·11-s + 126·12-s − 6·13-s − 42·14-s − 18·15-s + 126·16-s − 6·17-s + 126·18-s − 14·19-s − 63·20-s − 42·21-s − 48·22-s − 8·23-s + 336·24-s − 17·25-s − 36·26-s + 56·27-s − 147·28-s + ⋯
L(s)  = 1  + 4.24·2-s + 3.46·3-s + 21/2·4-s − 1.34·5-s + 14.6·6-s − 2.64·7-s + 19.7·8-s + 7·9-s − 5.69·10-s − 2.41·11-s + 36.3·12-s − 1.66·13-s − 11.2·14-s − 4.64·15-s + 63/2·16-s − 1.45·17-s + 29.6·18-s − 3.21·19-s − 14.0·20-s − 9.16·21-s − 10.2·22-s − 1.66·23-s + 68.5·24-s − 3.39·25-s − 7.06·26-s + 10.7·27-s − 27.7·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 17^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 17^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 17^{6} \cdot 59^{6}\)
Sign: $1$
Analytic conductor: \(1.23133\times 10^{10}\)
Root analytic conductor: \(6.93209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 17^{6} \cdot 59^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{6} \)
3 \( ( 1 - T )^{6} \)
17 \( ( 1 + T )^{6} \)
59 \( ( 1 + T )^{6} \)
good5 \( 1 + 3 T + 26 T^{2} + 63 T^{3} + 298 T^{4} + 576 T^{5} + 1928 T^{6} + 576 p T^{7} + 298 p^{2} T^{8} + 63 p^{3} T^{9} + 26 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + p T + 43 T^{2} + 176 T^{3} + 2 p^{3} T^{4} + 303 p T^{5} + 6264 T^{6} + 303 p^{2} T^{7} + 2 p^{5} T^{8} + 176 p^{3} T^{9} + 43 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} \)
11 \( 1 + 8 T + 64 T^{2} + 285 T^{3} + 1390 T^{4} + 4713 T^{5} + 18422 T^{6} + 4713 p T^{7} + 1390 p^{2} T^{8} + 285 p^{3} T^{9} + 64 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 6 T + 51 T^{2} + 202 T^{3} + 1168 T^{4} + 3626 T^{5} + 17130 T^{6} + 3626 p T^{7} + 1168 p^{2} T^{8} + 202 p^{3} T^{9} + 51 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 14 T + 142 T^{2} + 1032 T^{3} + 6430 T^{4} + 1770 p T^{5} + 158058 T^{6} + 1770 p^{2} T^{7} + 6430 p^{2} T^{8} + 1032 p^{3} T^{9} + 142 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 8 T + 58 T^{2} + 239 T^{3} + 1660 T^{4} + 5627 T^{5} + 33542 T^{6} + 5627 p T^{7} + 1660 p^{2} T^{8} + 239 p^{3} T^{9} + 58 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 21 T + 308 T^{2} + 3233 T^{3} + 27644 T^{4} + 192116 T^{5} + 1134652 T^{6} + 192116 p T^{7} + 27644 p^{2} T^{8} + 3233 p^{3} T^{9} + 308 p^{4} T^{10} + 21 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 5 T + 120 T^{2} + 335 T^{3} + 6220 T^{4} + 10152 T^{5} + 216224 T^{6} + 10152 p T^{7} + 6220 p^{2} T^{8} + 335 p^{3} T^{9} + 120 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 13 T + 174 T^{2} + 1079 T^{3} + 7478 T^{4} + 24414 T^{5} + 184720 T^{6} + 24414 p T^{7} + 7478 p^{2} T^{8} + 1079 p^{3} T^{9} + 174 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 18 T + 312 T^{2} + 83 p T^{3} + 34028 T^{4} + 263831 T^{5} + 1877422 T^{6} + 263831 p T^{7} + 34028 p^{2} T^{8} + 83 p^{4} T^{9} + 312 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 + 4 T + 82 T^{2} - 119 T^{3} + 3276 T^{4} - 7875 T^{5} + 196154 T^{6} - 7875 p T^{7} + 3276 p^{2} T^{8} - 119 p^{3} T^{9} + 82 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 4 T + 138 T^{2} + 619 T^{3} + 10402 T^{4} + 47159 T^{5} + 541662 T^{6} + 47159 p T^{7} + 10402 p^{2} T^{8} + 619 p^{3} T^{9} + 138 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 25 T + 485 T^{2} + 6416 T^{3} + 71862 T^{4} + 651131 T^{5} + 5175170 T^{6} + 651131 p T^{7} + 71862 p^{2} T^{8} + 6416 p^{3} T^{9} + 485 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 308 T^{2} + 129 T^{3} + 42246 T^{4} + 21563 T^{5} + 3317184 T^{6} + 21563 p T^{7} + 42246 p^{2} T^{8} + 129 p^{3} T^{9} + 308 p^{4} T^{10} + p^{6} T^{12} \)
67 \( 1 + 13 T + 353 T^{2} + 3480 T^{3} + 54244 T^{4} + 421823 T^{5} + 4719308 T^{6} + 421823 p T^{7} + 54244 p^{2} T^{8} + 3480 p^{3} T^{9} + 353 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 12 T + 208 T^{2} + 1643 T^{3} + 14798 T^{4} + 127067 T^{5} + 921380 T^{6} + 127067 p T^{7} + 14798 p^{2} T^{8} + 1643 p^{3} T^{9} + 208 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 4 T + 84 T^{2} - 1103 T^{3} + 3268 T^{4} - 32085 T^{5} + 1185158 T^{6} - 32085 p T^{7} + 3268 p^{2} T^{8} - 1103 p^{3} T^{9} + 84 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 12 T + 412 T^{2} + 3560 T^{3} + 70624 T^{4} + 471276 T^{5} + 7011018 T^{6} + 471276 p T^{7} + 70624 p^{2} T^{8} + 3560 p^{3} T^{9} + 412 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 9 T + 316 T^{2} - 3677 T^{3} + 48348 T^{4} - 599848 T^{5} + 4785622 T^{6} - 599848 p T^{7} + 48348 p^{2} T^{8} - 3677 p^{3} T^{9} + 316 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 11 T + 526 T^{2} + 4607 T^{3} + 115612 T^{4} + 794778 T^{5} + 13655926 T^{6} + 794778 p T^{7} + 115612 p^{2} T^{8} + 4607 p^{3} T^{9} + 526 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 16 T + 3 p T^{2} - 4292 T^{3} + 61432 T^{4} - 686420 T^{5} + 6799904 T^{6} - 686420 p T^{7} + 61432 p^{2} T^{8} - 4292 p^{3} T^{9} + 3 p^{5} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.35702216676201136464338814674, −4.21521302604426858044841083053, −4.02521335402705904517356592335, −4.01595987136681055146961382116, −3.95649200958191246237773073111, −3.83696052561000720474770121819, −3.82563453354281331858838946155, −3.52265342412230438634782543711, −3.43646673253742209061378651208, −3.28425707247060373880870941913, −3.25829110091485406179416293394, −3.19931978158229223853339151015, −3.16787938207061625887660750624, −2.63521841315043912642928360461, −2.61104968770698869773037125721, −2.55628191986725704637445220500, −2.49517112455505472614658020780, −2.43252700888277450418764627644, −2.13512025491712521838839710011, −2.00215973052101583884117576647, −1.72812162739170178230731661288, −1.67629135270976817652906575570, −1.64189032353907375250147281203, −1.58777900756675827738509875206, −1.56441127454342634526107652731, 0, 0, 0, 0, 0, 0, 1.56441127454342634526107652731, 1.58777900756675827738509875206, 1.64189032353907375250147281203, 1.67629135270976817652906575570, 1.72812162739170178230731661288, 2.00215973052101583884117576647, 2.13512025491712521838839710011, 2.43252700888277450418764627644, 2.49517112455505472614658020780, 2.55628191986725704637445220500, 2.61104968770698869773037125721, 2.63521841315043912642928360461, 3.16787938207061625887660750624, 3.19931978158229223853339151015, 3.25829110091485406179416293394, 3.28425707247060373880870941913, 3.43646673253742209061378651208, 3.52265342412230438634782543711, 3.82563453354281331858838946155, 3.83696052561000720474770121819, 3.95649200958191246237773073111, 4.01595987136681055146961382116, 4.02521335402705904517356592335, 4.21521302604426858044841083053, 4.35702216676201136464338814674

Graph of the $Z$-function along the critical line

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