Properties

Label 4-825e2-1.1-c1e2-0-18
Degree $4$
Conductor $680625$
Sign $1$
Analytic cond. $43.3972$
Root an. cond. $2.56664$
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  − 2·3-s − 4-s − 4·7-s + 3·9-s − 2·11-s + 2·12-s − 4·13-s − 3·16-s + 4·19-s + 8·21-s − 4·27-s + 4·28-s − 8·31-s + 4·33-s − 3·36-s − 4·37-s + 8·39-s − 4·43-s + 2·44-s + 6·48-s − 2·49-s + 4·52-s + 12·53-s − 8·57-s + 4·61-s − 12·63-s + 7·64-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 1.51·7-s + 9-s − 0.603·11-s + 0.577·12-s − 1.10·13-s − 3/4·16-s + 0.917·19-s + 1.74·21-s − 0.769·27-s + 0.755·28-s − 1.43·31-s + 0.696·33-s − 1/2·36-s − 0.657·37-s + 1.28·39-s − 0.609·43-s + 0.301·44-s + 0.866·48-s − 2/7·49-s + 0.554·52-s + 1.64·53-s − 1.05·57-s + 0.512·61-s − 1.51·63-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(680625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(43.3972\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 680625,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−10.03295306144857499421045595323, −9.686714701389686651026132086190, −9.370474085062089692376184232137, −8.861666555025288461035241335318, −8.419121648308962260695848604775, −7.72974972916437071272652813370, −7.15048228332180799808962322684, −7.01502307853168229988256702430, −6.68715608269639816024460523300, −5.89127471605424311936608355157, −5.46549594401565070013609588439, −5.42177654495458161759189796888, −4.44199002570797177639425616641, −4.43066430794526292534749169160, −3.49179412737773871035965089826, −3.04954229792981685633767489801, −2.37648288046462023441843942037, −1.41796143708679275545704644942, 0, 0, 1.41796143708679275545704644942, 2.37648288046462023441843942037, 3.04954229792981685633767489801, 3.49179412737773871035965089826, 4.43066430794526292534749169160, 4.44199002570797177639425616641, 5.42177654495458161759189796888, 5.46549594401565070013609588439, 5.89127471605424311936608355157, 6.68715608269639816024460523300, 7.01502307853168229988256702430, 7.15048228332180799808962322684, 7.72974972916437071272652813370, 8.419121648308962260695848604775, 8.861666555025288461035241335318, 9.370474085062089692376184232137, 9.686714701389686651026132086190, 10.03295306144857499421045595323

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