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Properties

Label 12-7600e6-1.1-c1e6-0-5
Degree $12$
Conductor $1.927\times 10^{23}$
Sign $1$
Analytic cond. $4.99510\times 10^{10}$
Root an. cond. $7.79014$
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  − 2·3-s − 2·7-s − 2·9-s − 3·11-s − 3·13-s − 2·17-s − 6·19-s + 4·21-s − 4·23-s + 2·27-s + 7·29-s − 5·31-s + 6·33-s + 6·39-s + 11·41-s + 7·43-s − 20·47-s − 20·49-s + 4·51-s − 7·53-s + 12·57-s + 4·59-s + 13·61-s + 4·63-s − 25·67-s + 8·69-s − 29·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s − 2/3·9-s − 0.904·11-s − 0.832·13-s − 0.485·17-s − 1.37·19-s + 0.872·21-s − 0.834·23-s + 0.384·27-s + 1.29·29-s − 0.898·31-s + 1.04·33-s + 0.960·39-s + 1.71·41-s + 1.06·43-s − 2.91·47-s − 2.85·49-s + 0.560·51-s − 0.961·53-s + 1.58·57-s + 0.520·59-s + 1.66·61-s + 0.503·63-s − 3.05·67-s + 0.963·69-s − 3.44·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 5^{12} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(4.99510\times 10^{10}\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{24} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

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 \( 1 \)
5 \( 1 \)
19 \( ( 1 + T )^{6} \)
good3 \( 1 + 2 T + 2 p T^{2} + 14 T^{3} + 8 p T^{4} + 40 T^{5} + 83 T^{6} + 40 p T^{7} + 8 p^{3} T^{8} + 14 p^{3} T^{9} + 2 p^{5} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 2 T + 24 T^{2} + 54 T^{3} + 318 T^{4} + 594 T^{5} + 2791 T^{6} + 594 p T^{7} + 318 p^{2} T^{8} + 54 p^{3} T^{9} + 24 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 3 T + 14 T^{2} + 50 T^{3} + 433 T^{4} + 915 T^{5} + 3400 T^{6} + 915 p T^{7} + 433 p^{2} T^{8} + 50 p^{3} T^{9} + 14 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 3 T + 35 T^{2} + 134 T^{3} + 790 T^{4} + 2388 T^{5} + 13141 T^{6} + 2388 p T^{7} + 790 p^{2} T^{8} + 134 p^{3} T^{9} + 35 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 2 T + 40 T^{2} - 12 T^{3} + 422 T^{4} - 2878 T^{5} + 309 T^{6} - 2878 p T^{7} + 422 p^{2} T^{8} - 12 p^{3} T^{9} + 40 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 4 T + 96 T^{2} + 346 T^{3} + 4540 T^{4} + 14008 T^{5} + 131019 T^{6} + 14008 p T^{7} + 4540 p^{2} T^{8} + 346 p^{3} T^{9} + 96 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 - 7 T + 101 T^{2} - 680 T^{3} + 6104 T^{4} - 32894 T^{5} + 218211 T^{6} - 32894 p T^{7} + 6104 p^{2} T^{8} - 680 p^{3} T^{9} + 101 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 5 T + 116 T^{2} + 166 T^{3} + 4207 T^{4} - 9815 T^{5} + 97168 T^{6} - 9815 p T^{7} + 4207 p^{2} T^{8} + 166 p^{3} T^{9} + 116 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 + 14 T^{2} + 189 T^{3} + 1675 T^{4} + 1758 T^{5} + 75767 T^{6} + 1758 p T^{7} + 1675 p^{2} T^{8} + 189 p^{3} T^{9} + 14 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 11 T + 162 T^{2} - 1616 T^{3} + 14575 T^{4} - 107273 T^{5} + 784764 T^{6} - 107273 p T^{7} + 14575 p^{2} T^{8} - 1616 p^{3} T^{9} + 162 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 7 T + p T^{2} - 530 T^{3} + 3441 T^{4} - 21767 T^{5} + 216350 T^{6} - 21767 p T^{7} + 3441 p^{2} T^{8} - 530 p^{3} T^{9} + p^{5} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 20 T + 294 T^{2} + 2363 T^{3} + 14371 T^{4} + 39728 T^{5} + 169611 T^{6} + 39728 p T^{7} + 14371 p^{2} T^{8} + 2363 p^{3} T^{9} + 294 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 7 T + 286 T^{2} + 1674 T^{3} + 35709 T^{4} + 168565 T^{5} + 2474973 T^{6} + 168565 p T^{7} + 35709 p^{2} T^{8} + 1674 p^{3} T^{9} + 286 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 4 T + 261 T^{2} - 825 T^{3} + 32193 T^{4} - 80901 T^{5} + 2380506 T^{6} - 80901 p T^{7} + 32193 p^{2} T^{8} - 825 p^{3} T^{9} + 261 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 13 T + 246 T^{2} - 2418 T^{3} + 29973 T^{4} - 235869 T^{5} + 2276056 T^{6} - 235869 p T^{7} + 29973 p^{2} T^{8} - 2418 p^{3} T^{9} + 246 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 25 T + 505 T^{2} + 6772 T^{3} + 80896 T^{4} + 773578 T^{5} + 6921907 T^{6} + 773578 p T^{7} + 80896 p^{2} T^{8} + 6772 p^{3} T^{9} + 505 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 29 T + 603 T^{2} + 8526 T^{3} + 102909 T^{4} + 1014029 T^{5} + 9209494 T^{6} + 1014029 p T^{7} + 102909 p^{2} T^{8} + 8526 p^{3} T^{9} + 603 p^{4} T^{10} + 29 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 19 T + 505 T^{2} + 6558 T^{3} + 98292 T^{4} + 932744 T^{5} + 9748099 T^{6} + 932744 p T^{7} + 98292 p^{2} T^{8} + 6558 p^{3} T^{9} + 505 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 28 T + 659 T^{2} + 10371 T^{3} + 145815 T^{4} + 1597403 T^{5} + 15814250 T^{6} + 1597403 p T^{7} + 145815 p^{2} T^{8} + 10371 p^{3} T^{9} + 659 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 15 T + 496 T^{2} - 5598 T^{3} + 100571 T^{4} - 875935 T^{5} + 11004272 T^{6} - 875935 p T^{7} + 100571 p^{2} T^{8} - 5598 p^{3} T^{9} + 496 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 12 T + 317 T^{2} + 3161 T^{3} + 56077 T^{4} + 457091 T^{5} + 6094922 T^{6} + 457091 p T^{7} + 56077 p^{2} T^{8} + 3161 p^{3} T^{9} + 317 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 13 T + 448 T^{2} + 3788 T^{3} + 80619 T^{4} + 494279 T^{5} + 9036752 T^{6} + 494279 p T^{7} + 80619 p^{2} T^{8} + 3788 p^{3} T^{9} + 448 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.36676476902041552046454200249, −4.36230357836642365054978148586, −4.09178084112276884074677868663, −4.03074601243493776864288979693, −3.93676183210762841659172416940, −3.70110372760700890770195139898, −3.68635604612165284776725692306, −3.48603336762707828547689857725, −3.31351080060677559934017819840, −3.09205691665947725491564995582, −2.97845914929348359928696872621, −2.87159552950521352379257621528, −2.85141467397530161618638254940, −2.49471826207213807371905269683, −2.45372036117674958744443515264, −2.42255912755887699568598329853, −2.35304463057718885774617324629, −2.23343770572331742068642760893, −1.67391833147898922553787888617, −1.55965799822884664057354508439, −1.51139022598239703566608084162, −1.40736898170938260515380454094, −1.28045491613866060849014307113, −1.09765912467188099419211378574, −0.73001629642707044811614098368, 0, 0, 0, 0, 0, 0, 0.73001629642707044811614098368, 1.09765912467188099419211378574, 1.28045491613866060849014307113, 1.40736898170938260515380454094, 1.51139022598239703566608084162, 1.55965799822884664057354508439, 1.67391833147898922553787888617, 2.23343770572331742068642760893, 2.35304463057718885774617324629, 2.42255912755887699568598329853, 2.45372036117674958744443515264, 2.49471826207213807371905269683, 2.85141467397530161618638254940, 2.87159552950521352379257621528, 2.97845914929348359928696872621, 3.09205691665947725491564995582, 3.31351080060677559934017819840, 3.48603336762707828547689857725, 3.68635604612165284776725692306, 3.70110372760700890770195139898, 3.93676183210762841659172416940, 4.03074601243493776864288979693, 4.09178084112276884074677868663, 4.36230357836642365054978148586, 4.36676476902041552046454200249

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

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