Properties

Degree 4
Conductor $ 5^{2} \cdot 31 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 2·5-s + 2·9-s + 5·16-s + 6·20-s − 25-s − 4·29-s + 7·31-s − 6·36-s − 12·41-s − 4·45-s + 2·49-s + 8·59-s + 4·61-s − 3·64-s − 8·71-s + 16·79-s − 10·80-s − 5·81-s + 4·89-s + 3·100-s − 4·101-s − 4·109-s + 12·116-s − 6·121-s − 21·124-s + 12·125-s + ⋯
L(s)  = 1  − 3/2·4-s − 0.894·5-s + 2/3·9-s + 5/4·16-s + 1.34·20-s − 1/5·25-s − 0.742·29-s + 1.25·31-s − 36-s − 1.87·41-s − 0.596·45-s + 2/7·49-s + 1.04·59-s + 0.512·61-s − 3/8·64-s − 0.949·71-s + 1.80·79-s − 1.11·80-s − 5/9·81-s + 0.423·89-s + 3/10·100-s − 0.398·101-s − 0.383·109-s + 1.11·116-s − 0.545·121-s − 1.88·124-s + 1.07·125-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(775\)    =    \(5^{2} \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{775} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 775,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.3599289594$
$L(\frac12)$  $\approx$  $0.3599289594$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;31\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;31\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_2$ \( 1 + 2 T + p T^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 8 T + p T^{2} ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−14.62453651579960752855609705993, −13.77869867733555408227140097541, −13.39532508842783225112744987792, −12.76959912763854130639727432235, −12.03048402116051023312789572862, −11.46251724435172038002889334381, −10.36980515604252195065498681887, −9.848375516162024926849324264990, −9.045045414591489573501636252557, −8.345116428302583235325818032228, −7.71014222902365690593838096253, −6.70285425315650878718056220114, −5.33448429556593353077760382433, −4.44825760168931727037771499842, −3.67899147579235780516804292863, 3.67899147579235780516804292863, 4.44825760168931727037771499842, 5.33448429556593353077760382433, 6.70285425315650878718056220114, 7.71014222902365690593838096253, 8.345116428302583235325818032228, 9.045045414591489573501636252557, 9.848375516162024926849324264990, 10.36980515604252195065498681887, 11.46251724435172038002889334381, 12.03048402116051023312789572862, 12.76959912763854130639727432235, 13.39532508842783225112744987792, 13.77869867733555408227140097541, 14.62453651579960752855609705993

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