Properties

Degree 4
Conductor $ 5^{2} \cdot 31^{2} $
Sign $1$
Motivic weight 0
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 5-s − 2·9-s + 2·19-s − 20-s − 2·31-s + 2·36-s − 2·41-s − 2·45-s − 49-s − 2·59-s + 64-s + 2·71-s − 2·76-s + 3·81-s + 2·95-s + 2·101-s + 2·109-s + 2·121-s + 2·124-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 4-s + 5-s − 2·9-s + 2·19-s − 20-s − 2·31-s + 2·36-s − 2·41-s − 2·45-s − 49-s − 2·59-s + 64-s + 2·71-s − 2·76-s + 3·81-s + 2·95-s + 2·101-s + 2·109-s + 2·121-s + 2·124-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 24025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(24025\)    =    \(5^{2} \cdot 31^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{155} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 24025,\ (\ :0, 0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.3694422763$
$L(\frac12)$  $\approx$  $0.3694422763$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−14.01728269487878139040519980875, −13.24129811164260988978612131150, −12.46543236559967768776943439337, −11.95316490223993661237857601247, −11.32697481286746022943537966114, −11.15479282252388540651046991477, −10.30377698260116748695414024712, −9.749128852540363752407603347420, −9.234470675715962152120679315755, −9.069309539962566150968174134441, −8.406564653898943386903774874381, −7.87127156377874814137322110235, −7.16240109251553910414852154377, −6.28352037519677019263430211854, −5.79099571878796810866265396472, −5.14454408149836053986851032894, −4.99559637066076860419486785128, −3.57720643177974959481399999539, −3.14553519709855962013572144934, −1.98470118185236666196526716097, 1.98470118185236666196526716097, 3.14553519709855962013572144934, 3.57720643177974959481399999539, 4.99559637066076860419486785128, 5.14454408149836053986851032894, 5.79099571878796810866265396472, 6.28352037519677019263430211854, 7.16240109251553910414852154377, 7.87127156377874814137322110235, 8.406564653898943386903774874381, 9.069309539962566150968174134441, 9.234470675715962152120679315755, 9.749128852540363752407603347420, 10.30377698260116748695414024712, 11.15479282252388540651046991477, 11.32697481286746022943537966114, 11.95316490223993661237857601247, 12.46543236559967768776943439337, 13.24129811164260988978612131150, 14.01728269487878139040519980875

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