Properties

Degree 4
Conductor $ 5^{2} \cdot 13^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4-s + 2·5-s − 8·8-s + 2·9-s + 4·10-s − 6·13-s − 7·16-s + 4·18-s − 2·20-s − 25-s − 12·26-s + 12·29-s + 14·32-s − 2·36-s − 12·37-s − 16·40-s + 4·45-s + 16·47-s − 14·49-s − 2·50-s + 6·52-s + 24·58-s + 12·61-s + 35·64-s − 12·65-s − 24·67-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s + 0.894·5-s − 2.82·8-s + 2/3·9-s + 1.26·10-s − 1.66·13-s − 7/4·16-s + 0.942·18-s − 0.447·20-s − 1/5·25-s − 2.35·26-s + 2.22·29-s + 2.47·32-s − 1/3·36-s − 1.97·37-s − 2.52·40-s + 0.596·45-s + 2.33·47-s − 2·49-s − 0.282·50-s + 0.832·52-s + 3.15·58-s + 1.53·61-s + 35/8·64-s − 1.48·65-s − 2.93·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4225\)    =    \(5^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4225} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4225,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.221798417$
$L(\frac12)$  $\approx$  $1.221798417$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{5,\;13\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 138 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.71714119571974731062634462663, −11.98239077020634614152784228637, −11.90311223456943156683204875337, −10.33586677104772337841262224795, −10.15382394013077472054980476610, −9.463934432947145288101362720902, −8.972634087180848766990541926844, −8.260808297600080955067953336526, −7.23591734839184107054042839302, −6.42372356969681330659811031444, −5.73514302994749178233236282911, −4.88156617889206626562591184346, −4.72173132163098801836255991991, −3.64189765661446866968181750927, −2.58431702142523155261267406595, 2.58431702142523155261267406595, 3.64189765661446866968181750927, 4.72173132163098801836255991991, 4.88156617889206626562591184346, 5.73514302994749178233236282911, 6.42372356969681330659811031444, 7.23591734839184107054042839302, 8.260808297600080955067953336526, 8.972634087180848766990541926844, 9.463934432947145288101362720902, 10.15382394013077472054980476610, 10.33586677104772337841262224795, 11.90311223456943156683204875337, 11.98239077020634614152784228637, 12.71714119571974731062634462663

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