Properties

Label 4-24e4-1.1-c1e2-0-33
Degree $4$
Conductor $331776$
Sign $-1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 3·9-s − 11-s − 13-s + 4·19-s + 5·23-s − 5·25-s + 3·29-s + 31-s + 35-s − 2·37-s + 5·41-s + 7·43-s + 3·45-s + 3·47-s − 9·49-s − 10·53-s + 55-s + 59-s + 7·61-s + 3·63-s + 65-s + 67-s − 4·73-s + 77-s − 9·79-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 9-s − 0.301·11-s − 0.277·13-s + 0.917·19-s + 1.04·23-s − 25-s + 0.557·29-s + 0.179·31-s + 0.169·35-s − 0.328·37-s + 0.780·41-s + 1.06·43-s + 0.447·45-s + 0.437·47-s − 9/7·49-s − 1.37·53-s + 0.134·55-s + 0.130·59-s + 0.896·61-s + 0.377·63-s + 0.124·65-s + 0.122·67-s − 0.468·73-s + 0.113·77-s − 1.01·79-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(21.1543\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ -1)\)

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$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
good5$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \)
29$D_{4}$ \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 7 T + 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 3 T + 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 7 T + 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 9 T + 18 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−13.0129905324, −12.8260427162, −12.3839317721, −11.9289575134, −11.4651634422, −11.2094375267, −10.8827944539, −10.2287992623, −9.78298573152, −9.47289504348, −9.00297163411, −8.38209551114, −8.22448377395, −7.51513758130, −7.27584825662, −6.72006927356, −5.99560115631, −5.80406221855, −5.13264588826, −4.69140786325, −4.02335752028, −3.34426514991, −2.91921953643, −2.34538166166, −1.18164442820, 0, 1.18164442820, 2.34538166166, 2.91921953643, 3.34426514991, 4.02335752028, 4.69140786325, 5.13264588826, 5.80406221855, 5.99560115631, 6.72006927356, 7.27584825662, 7.51513758130, 8.22448377395, 8.38209551114, 9.00297163411, 9.47289504348, 9.78298573152, 10.2287992623, 10.8827944539, 11.2094375267, 11.4651634422, 11.9289575134, 12.3839317721, 12.8260427162, 13.0129905324

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