Properties

Degree 4
Conductor $ 2 \cdot 113^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 4·7-s − 2·8-s − 4·9-s + 4·11-s + 4·13-s + 16-s − 6·25-s − 4·28-s + 12·31-s + 4·32-s + 4·36-s + 8·41-s − 4·44-s + 2·49-s − 4·52-s + 4·53-s − 8·56-s − 12·61-s − 16·63-s + 3·64-s + 8·72-s + 16·77-s + 7·81-s + 12·83-s − 8·88-s + 16·91-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.51·7-s − 0.707·8-s − 4/3·9-s + 1.20·11-s + 1.10·13-s + 1/4·16-s − 6/5·25-s − 0.755·28-s + 2.15·31-s + 0.707·32-s + 2/3·36-s + 1.24·41-s − 0.603·44-s + 2/7·49-s − 0.554·52-s + 0.549·53-s − 1.06·56-s − 1.53·61-s − 2.01·63-s + 3/8·64-s + 0.942·72-s + 1.82·77-s + 7/9·81-s + 1.31·83-s − 0.852·88-s + 1.67·91-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 25538 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 25538 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(25538\)    =    \(2 \cdot 113^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{25538} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 25538,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.195737336$
$L(\frac12)$  $\approx$  $1.195737336$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;113\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;113\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
113$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
43$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
59$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 100 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
89$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−10.99339513567532214010358711057, −10.03738908888083438005343610998, −9.482739262741147690691968106192, −8.920098062622682066518846558395, −8.522612671281386109551214519313, −8.143959498729516225099390774301, −7.66564172593200134495041605676, −6.55365565146131181831641349725, −6.09300940768430969280589724005, −5.67610659684221795644477219111, −4.82704049962210373706393517645, −4.22376300887116086081841179410, −3.49656800984114260862676677094, −2.53977820924588088282769973173, −1.25138314974718007440597741029, 1.25138314974718007440597741029, 2.53977820924588088282769973173, 3.49656800984114260862676677094, 4.22376300887116086081841179410, 4.82704049962210373706393517645, 5.67610659684221795644477219111, 6.09300940768430969280589724005, 6.55365565146131181831641349725, 7.66564172593200134495041605676, 8.143959498729516225099390774301, 8.522612671281386109551214519313, 8.920098062622682066518846558395, 9.482739262741147690691968106192, 10.03738908888083438005343610998, 10.99339513567532214010358711057

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