Properties

Label 2-30960-1.1-c1-0-14
Degree $2$
Conductor $30960$
Sign $1$
Analytic cond. $247.216$
Root an. cond. $15.7231$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 5·11-s + 13-s − 5·17-s − 4·19-s + 3·23-s + 25-s + 8·29-s + 9·31-s + 8·37-s + 5·41-s + 43-s + 8·47-s − 7·49-s + 13·53-s − 5·55-s + 8·59-s − 8·61-s − 65-s − 3·67-s − 8·71-s + 4·73-s − 9·83-s + 5·85-s − 12·89-s + 4·95-s + 5·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.50·11-s + 0.277·13-s − 1.21·17-s − 0.917·19-s + 0.625·23-s + 1/5·25-s + 1.48·29-s + 1.61·31-s + 1.31·37-s + 0.780·41-s + 0.152·43-s + 1.16·47-s − 49-s + 1.78·53-s − 0.674·55-s + 1.04·59-s − 1.02·61-s − 0.124·65-s − 0.366·67-s − 0.949·71-s + 0.468·73-s − 0.987·83-s + 0.542·85-s − 1.27·89-s + 0.410·95-s + 0.507·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30960\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $1$
Analytic conductor: \(247.216\)
Root analytic conductor: \(15.7231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{30960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 30960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.524498925\)
\(L(\frac12)\) \(\approx\) \(2.524498925\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
43 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.05179926578867, −14.62768222633607, −14.11750410883171, −13.42708295488683, −13.11405441175383, −12.27057667309507, −11.95648970364914, −11.38879220439335, −10.90799424651124, −10.35061653312198, −9.641470020923263, −9.054878394864516, −8.561482479950734, −8.224039749289192, −7.277734381078107, −6.763526108558472, −6.334534188752602, −5.802600346861575, −4.621545391628953, −4.395507044785718, −3.878695745201328, −2.893555645259988, −2.372434084283553, −1.290802306965514, −0.6760609821304975, 0.6760609821304975, 1.290802306965514, 2.372434084283553, 2.893555645259988, 3.878695745201328, 4.395507044785718, 4.621545391628953, 5.802600346861575, 6.334534188752602, 6.763526108558472, 7.277734381078107, 8.224039749289192, 8.561482479950734, 9.054878394864516, 9.641470020923263, 10.35061653312198, 10.90799424651124, 11.38879220439335, 11.95648970364914, 12.27057667309507, 13.11405441175383, 13.42708295488683, 14.11750410883171, 14.62768222633607, 15.05179926578867

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