Properties

Degree $4$
Conductor $25921$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 2·4-s − 2·5-s + 2·6-s − 2·7-s + 3·8-s − 3·9-s + 2·10-s + 4·12-s − 4·13-s + 2·14-s + 4·15-s + 16-s + 3·18-s − 10·19-s + 4·20-s + 4·21-s − 2·23-s − 6·24-s − 2·25-s + 4·26-s + 14·27-s + 4·28-s + 6·29-s − 4·30-s − 18·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 4-s − 0.894·5-s + 0.816·6-s − 0.755·7-s + 1.06·8-s − 9-s + 0.632·10-s + 1.15·12-s − 1.10·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s + 0.707·18-s − 2.29·19-s + 0.894·20-s + 0.872·21-s − 0.417·23-s − 1.22·24-s − 2/5·25-s + 0.784·26-s + 2.69·27-s + 0.755·28-s + 1.11·29-s − 0.730·30-s − 3.23·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25921\)    =    \(7^{2} \cdot 23^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{161} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 25921,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad7$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 201 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 101 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 178 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
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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

−12.28727604748612677389813916111, −12.21797440548279025324990261865, −11.70087467236819340133486872161, −10.97500514315244596432963183965, −10.64045259149951998424579849140, −10.25401701706821133107508669339, −9.458195711122420591134227549652, −9.064200128179016075487121723875, −8.413407376785878903737385167234, −8.331537122973244962504931509747, −7.33816763666406712710264269952, −6.90236653477683376803143602792, −5.94195185101856910183950290577, −5.74648271268377232924353020204, −4.82721009424870334331669025534, −4.33032702653995820299645518164, −3.57697056385459043134029716675, −2.44933117251314136763665246411, 0, 0, 2.44933117251314136763665246411, 3.57697056385459043134029716675, 4.33032702653995820299645518164, 4.82721009424870334331669025534, 5.74648271268377232924353020204, 5.94195185101856910183950290577, 6.90236653477683376803143602792, 7.33816763666406712710264269952, 8.331537122973244962504931509747, 8.413407376785878903737385167234, 9.064200128179016075487121723875, 9.458195711122420591134227549652, 10.25401701706821133107508669339, 10.64045259149951998424579849140, 10.97500514315244596432963183965, 11.70087467236819340133486872161, 12.21797440548279025324990261865, 12.28727604748612677389813916111

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