Properties

Label 4-585e2-1.1-c1e2-0-19
Degree $4$
Conductor $342225$
Sign $1$
Analytic cond. $21.8205$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 2·4-s − 5-s + 4·7-s + 6·9-s + 6·12-s − 13-s − 3·15-s − 6·17-s + 4·19-s − 2·20-s + 12·21-s − 3·23-s + 9·27-s + 8·28-s − 6·29-s + 4·31-s − 4·35-s + 12·36-s + 16·37-s − 3·39-s + 12·41-s + 43-s − 6·45-s + 7·49-s − 18·51-s − 2·52-s + ⋯
L(s)  = 1  + 1.73·3-s + 4-s − 0.447·5-s + 1.51·7-s + 2·9-s + 1.73·12-s − 0.277·13-s − 0.774·15-s − 1.45·17-s + 0.917·19-s − 0.447·20-s + 2.61·21-s − 0.625·23-s + 1.73·27-s + 1.51·28-s − 1.11·29-s + 0.718·31-s − 0.676·35-s + 2·36-s + 2.63·37-s − 0.480·39-s + 1.87·41-s + 0.152·43-s − 0.894·45-s + 49-s − 2.52·51-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(342225\)    =    \(3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(21.8205\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{585} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 342225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.887577639\)
\(L(\frac12)\) \(\approx\) \(4.887577639\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
13$C_2$ \( 1 + T + T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−10.93010050190578131709217144376, −10.78109804584532360233798481237, −9.689129838010136175015076660757, −9.576769722926216955152214017708, −9.272857555949568460027828887445, −8.531114317761551315792798865620, −8.187721209594576722270502240898, −7.87536326563363653993875262780, −7.40009806758018851346513081487, −7.32942057489335448414199797032, −6.46801139844877413932153507920, −6.13053430042132171226545804344, −5.26508227904351933532283758267, −4.62121900128406685981824770763, −4.20687796837548792967808654514, −3.84824401142287842985131840707, −2.70857127280437685586188788549, −2.65663678564189498846180835172, −1.97082098274164565420682191991, −1.30735363288778794687760949473, 1.30735363288778794687760949473, 1.97082098274164565420682191991, 2.65663678564189498846180835172, 2.70857127280437685586188788549, 3.84824401142287842985131840707, 4.20687796837548792967808654514, 4.62121900128406685981824770763, 5.26508227904351933532283758267, 6.13053430042132171226545804344, 6.46801139844877413932153507920, 7.32942057489335448414199797032, 7.40009806758018851346513081487, 7.87536326563363653993875262780, 8.187721209594576722270502240898, 8.531114317761551315792798865620, 9.272857555949568460027828887445, 9.576769722926216955152214017708, 9.689129838010136175015076660757, 10.78109804584532360233798481237, 10.93010050190578131709217144376

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