Properties

Label 4-1568e2-1.1-c1e2-0-34
Degree $4$
Conductor $2458624$
Sign $1$
Analytic cond. $156.763$
Root an. cond. $3.53843$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s − 8·25-s − 8·29-s − 24·37-s − 28·53-s + 27·81-s − 40·109-s + 28·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2·9-s − 8/5·25-s − 1.48·29-s − 3.94·37-s − 3.84·53-s + 3·81-s − 3.83·109-s + 2.63·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(156.763\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1568} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2458624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 120 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 96 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.066649967251363745631663814186, −8.940696255799733527976757968662, −8.392721858069374507578479998662, −8.144234687068523397466885463802, −7.64924418530679762834518186282, −7.39657363069087174600749246313, −6.58910494162729178610576965446, −6.50587711035816312129719493699, −5.78125465734212759409588014590, −5.67525630074849547285919795819, −4.99938843313010453410834149148, −4.98193085026202043096987197813, −3.92659363538105101606296112505, −3.63974803479058916205388777857, −3.19073958456411393365617171216, −2.71633622188400856576982296269, −1.89988458037923500135026192273, −1.64682263510376091891555248700, 0, 0, 1.64682263510376091891555248700, 1.89988458037923500135026192273, 2.71633622188400856576982296269, 3.19073958456411393365617171216, 3.63974803479058916205388777857, 3.92659363538105101606296112505, 4.98193085026202043096987197813, 4.99938843313010453410834149148, 5.67525630074849547285919795819, 5.78125465734212759409588014590, 6.50587711035816312129719493699, 6.58910494162729178610576965446, 7.39657363069087174600749246313, 7.64924418530679762834518186282, 8.144234687068523397466885463802, 8.392721858069374507578479998662, 8.940696255799733527976757968662, 9.066649967251363745631663814186

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