Properties

Degree 12
Conductor $ 2^{18} \cdot 3^{18} \cdot 7^{6} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s − 11-s + 4·13-s − 2·17-s + 5·19-s + 2·23-s + 11·25-s + 2·29-s − 4·31-s − 4·35-s + 8·37-s − 6·43-s − 3·47-s + 2·49-s + 8·53-s − 55-s + 9·59-s + 13·61-s + 4·65-s + 16·67-s − 2·71-s − 3·73-s + 4·77-s + 12·79-s + 2·83-s − 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 0.301·11-s + 1.10·13-s − 0.485·17-s + 1.14·19-s + 0.417·23-s + 11/5·25-s + 0.371·29-s − 0.718·31-s − 0.676·35-s + 1.31·37-s − 0.914·43-s − 0.437·47-s + 2/7·49-s + 1.09·53-s − 0.134·55-s + 1.17·59-s + 1.66·61-s + 0.496·65-s + 1.95·67-s − 0.237·71-s − 0.351·73-s + 0.455·77-s + 1.35·79-s + 0.219·83-s − 0.216·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(2^{18} \cdot 3^{18} \cdot 7^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1512} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((12,\ 2^{18} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(9.361359788\)
\(L(\frac12)\)  \(\approx\)  \(9.361359788\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + 4 T + 2 p T^{2} + 55 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
good5 \( 1 - T - 2 p T^{2} + 7 T^{3} + 57 T^{4} - 14 T^{5} - 299 T^{6} - 14 p T^{7} + 57 p^{2} T^{8} + 7 p^{3} T^{9} - 2 p^{5} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + T - 20 T^{2} - 41 T^{3} + 179 T^{4} + 310 T^{5} - 1349 T^{6} + 310 p T^{7} + 179 p^{2} T^{8} - 41 p^{3} T^{9} - 20 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 - 2 T + 36 T^{2} - 49 T^{3} + 36 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 2 T - 36 T^{2} - 14 T^{3} + 826 T^{4} - 262 T^{5} - 16321 T^{6} - 262 p T^{7} + 826 p^{2} T^{8} - 14 p^{3} T^{9} - 36 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 5 T - 34 T^{2} + 63 T^{3} + 1465 T^{4} - 1156 T^{5} - 29405 T^{6} - 1156 p T^{7} + 1465 p^{2} T^{8} + 63 p^{3} T^{9} - 34 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 - 2 T - 60 T^{2} + 38 T^{3} + 2458 T^{4} - 482 T^{5} - 65101 T^{6} - 482 p T^{7} + 2458 p^{2} T^{8} + 38 p^{3} T^{9} - 60 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - T + 37 T^{2} + 71 T^{3} + 37 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( 1 + 4 T - 28 T^{2} - 402 T^{3} - 584 T^{4} + 5648 T^{5} + 67711 T^{6} + 5648 p T^{7} - 584 p^{2} T^{8} - 402 p^{3} T^{9} - 28 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 8 T - 2 T^{2} + 254 T^{3} - 1214 T^{4} + 2314 T^{5} + 5399 T^{6} + 2314 p T^{7} - 1214 p^{2} T^{8} + 254 p^{3} T^{9} - 2 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 66 T^{2} + 137 T^{3} + 66 p T^{4} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 3 T + 9 T^{2} - 143 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 3 T - 18 T^{2} - 1225 T^{3} - 2499 T^{4} + 16644 T^{5} + 579911 T^{6} + 16644 p T^{7} - 2499 p^{2} T^{8} - 1225 p^{3} T^{9} - 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 8 T - 90 T^{2} + 278 T^{3} + 9514 T^{4} - 8150 T^{5} - 568945 T^{6} - 8150 p T^{7} + 9514 p^{2} T^{8} + 278 p^{3} T^{9} - 90 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 9 T - 54 T^{2} + 171 T^{3} + 4023 T^{4} + 18486 T^{5} - 466229 T^{6} + 18486 p T^{7} + 4023 p^{2} T^{8} + 171 p^{3} T^{9} - 54 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 13 T - 45 T^{2} + 252 T^{3} + 13021 T^{4} - 33607 T^{5} - 667154 T^{6} - 33607 p T^{7} + 13021 p^{2} T^{8} + 252 p^{3} T^{9} - 45 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 16 T + 34 T^{2} + 562 T^{3} + 1498 T^{4} - 52510 T^{5} + 381563 T^{6} - 52510 p T^{7} + 1498 p^{2} T^{8} + 562 p^{3} T^{9} + 34 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 + T + 129 T^{2} + 235 T^{3} + 129 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 3 T - 132 T^{2} - 347 T^{3} + 8433 T^{4} + 8514 T^{5} - 573015 T^{6} + 8514 p T^{7} + 8433 p^{2} T^{8} - 347 p^{3} T^{9} - 132 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 12 T - 84 T^{2} + 454 T^{3} + 14832 T^{4} - 6264 T^{5} - 1475337 T^{6} - 6264 p T^{7} + 14832 p^{2} T^{8} + 454 p^{3} T^{9} - 84 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - T + 129 T^{2} - 313 T^{3} + 129 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 + 17 T - 57 T^{2} - 344 T^{3} + 36001 T^{4} + 164759 T^{5} - 1843186 T^{6} + 164759 p T^{7} + 36001 p^{2} T^{8} - 344 p^{3} T^{9} - 57 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 - 14 T + 331 T^{2} - 2740 T^{3} + 331 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.92076073893140022981632368810, −4.84933975492053464254357146098, −4.84275919853703934889760582857, −4.44024066074217019992901117038, −4.40652218884129334800096302880, −4.17431520608749698858982480075, −4.05737355081620069039807099690, −3.61843871458610222786406055279, −3.57864529782449982388119905103, −3.49306160634205942151300454286, −3.35336153208492440410585512588, −3.35216423536734408374929232604, −3.10250844305523989285314385760, −2.72184468797916047512288207293, −2.55526998090462977539079991872, −2.52776863876765964664147453265, −2.48549531242273983748690933330, −2.00147238675535366587906015662, −1.78267575912904134625600350754, −1.71387728531895547907486711531, −1.45519481291397627519341149704, −0.961122134908316418280108868256, −0.65249653878592473157851141255, −0.64806325990727449707328948672, −0.60557490849267730427019892201, 0.60557490849267730427019892201, 0.64806325990727449707328948672, 0.65249653878592473157851141255, 0.961122134908316418280108868256, 1.45519481291397627519341149704, 1.71387728531895547907486711531, 1.78267575912904134625600350754, 2.00147238675535366587906015662, 2.48549531242273983748690933330, 2.52776863876765964664147453265, 2.55526998090462977539079991872, 2.72184468797916047512288207293, 3.10250844305523989285314385760, 3.35216423536734408374929232604, 3.35336153208492440410585512588, 3.49306160634205942151300454286, 3.57864529782449982388119905103, 3.61843871458610222786406055279, 4.05737355081620069039807099690, 4.17431520608749698858982480075, 4.40652218884129334800096302880, 4.44024066074217019992901117038, 4.84275919853703934889760582857, 4.84933975492053464254357146098, 4.92076073893140022981632368810

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

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