Properties

Label 4-24e4-1.1-c6e2-0-12
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $17559.2$
Root an. cond. $11.5113$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 968·7-s − 6.73e3·13-s + 1.14e4·19-s + 992·25-s + 7.95e4·31-s − 1.05e5·37-s + 7.60e3·43-s + 4.67e5·49-s − 2.65e4·61-s + 3.37e5·67-s + 4.72e5·73-s + 7.02e4·79-s − 6.52e6·91-s − 6.42e5·97-s − 3.98e6·103-s − 3.88e5·109-s + 1.74e6·121-s + 127-s + 131-s + 1.11e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 2.82·7-s − 3.06·13-s + 1.67·19-s + 0.0634·25-s + 2.67·31-s − 2.07·37-s + 0.0955·43-s + 3.97·49-s − 0.116·61-s + 1.12·67-s + 1.21·73-s + 0.142·79-s − 8.65·91-s − 0.704·97-s − 3.64·103-s − 0.300·109-s + 0.985·121-s + 4.72·133-s + 5.05·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(17559.2\)
Root analytic conductor: \(11.5113\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{576} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.564566590\)
\(L(\frac12)\) \(\approx\) \(3.564566590\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 992 T^{2} + p^{12} T^{4} \)
7$C_2$ \( ( 1 - 484 T + p^{6} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 1745714 T^{2} + p^{12} T^{4} \)
13$C_2$ \( ( 1 + 3368 T + p^{6} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 48274976 T^{2} + p^{12} T^{4} \)
19$C_2$ \( ( 1 - 5744 T + p^{6} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 284666690 T^{2} + p^{12} T^{4} \)
29$C_2^2$ \( 1 - 327940544 T^{2} + p^{12} T^{4} \)
31$C_2$ \( ( 1 - 39796 T + p^{6} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 52526 T + p^{6} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 8128061984 T^{2} + p^{12} T^{4} \)
43$C_2$ \( ( 1 - 3800 T + p^{6} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 15661450658 T^{2} + p^{12} T^{4} \)
53$C_2^2$ \( 1 + 12666935680 T^{2} + p^{12} T^{4} \)
59$C_2^2$ \( 1 - 21940729490 T^{2} + p^{12} T^{4} \)
61$C_2$ \( ( 1 + 13250 T + p^{6} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 168968 T + p^{6} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 26256724990 T^{2} + p^{12} T^{4} \)
73$C_2$ \( ( 1 - 236144 T + p^{6} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 35116 T + p^{6} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 653760187346 T^{2} + p^{12} T^{4} \)
89$C_2^2$ \( 1 - 977236742720 T^{2} + p^{12} T^{4} \)
97$C_2$ \( ( 1 + 321424 T + p^{6} T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.35022171726521897036781096684, −9.543951896866464836713259394027, −9.213317446180268905437375063074, −8.435574006494326632353156161892, −8.098031324176693798511009557961, −7.86546637511547730109050287042, −7.44934367639735212854745379499, −6.97745736850846790714074030286, −6.61276169954335565554319859192, −5.45123449337657301546702552530, −5.24583512857765705917418361888, −5.00064836253169559777266452057, −4.59737755004628403176798868479, −4.14766820006059447331224570061, −3.19666628802339647221024496809, −2.46850392260194968053101770445, −2.31743698091971696467434712122, −1.48884625047421493810732870659, −1.16467125975890339456892349164, −0.38529768320131408649471333716, 0.38529768320131408649471333716, 1.16467125975890339456892349164, 1.48884625047421493810732870659, 2.31743698091971696467434712122, 2.46850392260194968053101770445, 3.19666628802339647221024496809, 4.14766820006059447331224570061, 4.59737755004628403176798868479, 5.00064836253169559777266452057, 5.24583512857765705917418361888, 5.45123449337657301546702552530, 6.61276169954335565554319859192, 6.97745736850846790714074030286, 7.44934367639735212854745379499, 7.86546637511547730109050287042, 8.098031324176693798511009557961, 8.435574006494326632353156161892, 9.213317446180268905437375063074, 9.543951896866464836713259394027, 10.35022171726521897036781096684

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