Properties

Degree 4
Conductor $ 31^{2} $
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  + 2-s − 2·3-s − 2·4-s + 2·5-s − 2·6-s − 4·7-s − 3·8-s + 2·9-s + 2·10-s + 4·11-s + 4·12-s − 2·13-s − 4·14-s − 4·15-s + 16-s + 6·17-s + 2·18-s − 4·20-s + 8·21-s + 4·22-s − 2·23-s + 6·24-s − 7·25-s − 2·26-s − 6·27-s + 8·28-s + 10·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 4-s + 0.894·5-s − 0.816·6-s − 1.51·7-s − 1.06·8-s + 2/3·9-s + 0.632·10-s + 1.20·11-s + 1.15·12-s − 0.554·13-s − 1.06·14-s − 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.471·18-s − 0.894·20-s + 1.74·21-s + 0.852·22-s − 0.417·23-s + 1.22·24-s − 7/5·25-s − 0.392·26-s − 1.15·27-s + 1.51·28-s + 1.85·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(961\)    =    \(31^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{961} (1, \cdot )$
Sato-Tate  :  $G_{3,3}$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 961,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4492877238$
$L(\frac12)$  $\approx$  $0.4492877238$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 31$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 31$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad31$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
3$V_4$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$V_4$ \( 1 + 33 T^{2} + p^{2} T^{4} \)
23$V_4$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$V_4$ \( 1 + 113 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 6 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 14 T + 163 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.877551788, −19.3848350149, −19.040359127, −18.2399794693, −17.6655545569, −17.15432509, −17.0360677253, −15.9909759648, −15.7940237524, −14.4824607646, −14.2050657597, −13.6432074962, −13.0106085677, −12.2780393671, −12.1941126764, −11.1843545725, −9.98888457416, −9.69468104839, −9.39276405152, −8.06687354405, −6.7517465125, −6.03278399652, −5.63181617198, −4.52821779105, −3.50551568172, 3.50551568172, 4.52821779105, 5.63181617198, 6.03278399652, 6.7517465125, 8.06687354405, 9.39276405152, 9.69468104839, 9.98888457416, 11.1843545725, 12.1941126764, 12.2780393671, 13.0106085677, 13.6432074962, 14.2050657597, 14.4824607646, 15.7940237524, 15.9909759648, 17.0360677253, 17.15432509, 17.6655545569, 18.2399794693, 19.040359127, 19.3848350149, 19.877551788

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