Properties

Degree 12
Conductor $ 3^{12} \cdot 7^{6} \cdot 11^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 6·7-s + 4·13-s + 8·16-s − 16·25-s − 24·28-s + 4·31-s + 8·37-s + 20·43-s + 21·49-s − 16·52-s + 16·61-s − 15·64-s + 24·67-s + 44·73-s + 8·79-s + 24·91-s − 16·97-s + 64·100-s − 8·103-s + 20·109-s + 48·112-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2·4-s + 2.26·7-s + 1.10·13-s + 2·16-s − 3.19·25-s − 4.53·28-s + 0.718·31-s + 1.31·37-s + 3.04·43-s + 3·49-s − 2.21·52-s + 2.04·61-s − 1.87·64-s + 2.93·67-s + 5.14·73-s + 0.900·79-s + 2.51·91-s − 1.62·97-s + 32/5·100-s − 0.788·103-s + 1.91·109-s + 4.53·112-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(3^{12} \cdot 7^{6} \cdot 11^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{7623} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(12,\ 3^{12} \cdot 7^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )$
$L(1)$  $\approx$  $13.93550124$
$L(\frac12)$  $\approx$  $13.93550124$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( ( 1 - T )^{6} \)
11 \( 1 \)
good2 \( 1 + p^{2} T^{2} + p^{3} T^{4} + 15 T^{6} + p^{5} T^{8} + p^{6} T^{10} + p^{6} T^{12} \)
5 \( 1 + 16 T^{2} + 28 p T^{4} + 834 T^{6} + 28 p^{3} T^{8} + 16 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 - 2 T + 20 T^{2} - 48 T^{3} + 20 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 54 T^{2} + 1199 T^{4} + 19316 T^{6} + 1199 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 + 20 T^{2} - 16 T^{3} + 20 p T^{4} + p^{3} T^{6} )^{2} \)
23 \( 1 + 30 T^{2} + 767 T^{4} + 11492 T^{6} + 767 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 52 T^{2} + 2892 T^{4} + 86614 T^{6} + 2892 p^{2} T^{8} + 52 p^{4} T^{10} + p^{6} T^{12} \)
31 \( ( 1 - 2 T + 17 T^{2} + 180 T^{3} + 17 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 4 T + 96 T^{2} - 262 T^{3} + 96 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( 1 + 74 T^{2} + 4479 T^{4} + 233228 T^{6} + 4479 p^{2} T^{8} + 74 p^{4} T^{10} + p^{6} T^{12} \)
43 \( ( 1 - 10 T + 125 T^{2} - 828 T^{3} + 125 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 + 124 T^{2} + 11060 T^{4} + 591870 T^{6} + 11060 p^{2} T^{8} + 124 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 + 98 T^{2} + 8247 T^{4} + 518972 T^{6} + 8247 p^{2} T^{8} + 98 p^{4} T^{10} + p^{6} T^{12} \)
59 \( 1 + 116 T^{2} + 6924 T^{4} + 354662 T^{6} + 6924 p^{2} T^{8} + 116 p^{4} T^{10} + p^{6} T^{12} \)
61 \( ( 1 - 8 T + 123 T^{2} - 704 T^{3} + 123 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - 12 T + 212 T^{2} - 1540 T^{3} + 212 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( 1 + 18 T^{2} + 6671 T^{4} - 156772 T^{6} + 6671 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} \)
73 \( ( 1 - 22 T + 360 T^{2} - 3448 T^{3} + 360 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
79 \( ( 1 - 4 T + 93 T^{2} + 8 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 + 262 T^{2} + 27815 T^{4} + 2144628 T^{6} + 27815 p^{2} T^{8} + 262 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 + 430 T^{2} + 85071 T^{4} + 9709540 T^{6} + 85071 p^{2} T^{8} + 430 p^{4} T^{10} + p^{6} T^{12} \)
97 \( ( 1 + 8 T + 235 T^{2} + 1112 T^{3} + 235 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−4.16690538738535255161732506568, −3.79614801481783340973314293714, −3.75595522709663940746520290204, −3.69675929417969286816618255226, −3.64818159727377542555943145031, −3.50057871878516180615734144846, −3.27501015491609553429871422532, −3.21784671379191344228315577512, −2.75870396242534950895593793844, −2.74228934061646963522479251641, −2.50741391368027020748183598654, −2.50275006919990130670322368239, −2.19879209197845954973301645805, −2.16585364846422136126328030950, −2.11361236339331091022617459059, −1.87168498146448790186960305177, −1.69313217360840606079657056050, −1.50271498224426775840810308586, −1.32092205019387171112824536178, −1.19886881378098332071349691802, −0.802716779898837679115979525407, −0.69309319429416206093615435197, −0.64900915382474614952383272758, −0.61698247248424901507282411809, −0.29404026559765735195646324177, 0.29404026559765735195646324177, 0.61698247248424901507282411809, 0.64900915382474614952383272758, 0.69309319429416206093615435197, 0.802716779898837679115979525407, 1.19886881378098332071349691802, 1.32092205019387171112824536178, 1.50271498224426775840810308586, 1.69313217360840606079657056050, 1.87168498146448790186960305177, 2.11361236339331091022617459059, 2.16585364846422136126328030950, 2.19879209197845954973301645805, 2.50275006919990130670322368239, 2.50741391368027020748183598654, 2.74228934061646963522479251641, 2.75870396242534950895593793844, 3.21784671379191344228315577512, 3.27501015491609553429871422532, 3.50057871878516180615734144846, 3.64818159727377542555943145031, 3.69675929417969286816618255226, 3.75595522709663940746520290204, 3.79614801481783340973314293714, 4.16690538738535255161732506568

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

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