Properties

Label 4-464-1.1-c1e2-0-0
Degree $4$
Conductor $464$
Sign $1$
Analytic cond. $0.0295850$
Root an. cond. $0.414732$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s − 5-s + 6-s − 2·7-s + 3·8-s − 9-s + 10-s + 5·11-s + 12-s − 3·13-s + 2·14-s + 15-s − 16-s + 2·17-s + 18-s − 4·19-s + 20-s + 2·21-s − 5·22-s + 4·23-s − 3·24-s + 25-s + 3·26-s + 2·28-s − 3·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s + 1.06·8-s − 1/3·9-s + 0.316·10-s + 1.50·11-s + 0.288·12-s − 0.832·13-s + 0.534·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.436·21-s − 1.06·22-s + 0.834·23-s − 0.612·24-s + 1/5·25-s + 0.588·26-s + 0.377·28-s − 0.557·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $1$
Analytic conductor: \(0.0295850\)
Root analytic conductor: \(0.414732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2253348567\)
\(L(\frac12)\) \(\approx\) \(0.2253348567\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
29$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$D_{4}$ \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$D_{4}$ \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 7 T + 14 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$D_{4}$ \( 1 + 2 T - 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−19.7799763686, −19.6425679287, −19.0659965038, −18.6299007505, −17.6946681957, −17.1652633229, −16.8307886784, −16.4408888110, −15.4386344415, −14.6484797689, −14.2333874898, −13.2211520456, −12.6078884480, −11.9022154514, −11.2015511979, −10.4036253514, −9.53161908133, −9.09328098742, −8.15843756838, −7.14828701178, −6.31162447897, −5.00731259437, −3.78898435203, 3.78898435203, 5.00731259437, 6.31162447897, 7.14828701178, 8.15843756838, 9.09328098742, 9.53161908133, 10.4036253514, 11.2015511979, 11.9022154514, 12.6078884480, 13.2211520456, 14.2333874898, 14.6484797689, 15.4386344415, 16.4408888110, 16.8307886784, 17.1652633229, 17.6946681957, 18.6299007505, 19.0659965038, 19.6425679287, 19.7799763686

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