Properties

Label 2-30960-1.1-c1-0-10
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 − 2·7-s + 2·13-s + 6·17-s − 8·19-s + 6·23-s + 25-s + 6·29-s + 4·31-s − 2·35-s + 8·37-s − 6·41-s − 43-s + 6·47-s − 3·49-s + 12·53-s − 4·61-s + 2·65-s + 4·67-s − 10·73-s − 8·79-s + 6·85-s − 6·89-s − 4·91-s − 8·95-s + 14·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.554·13-s + 1.45·17-s − 1.83·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.338·35-s + 1.31·37-s − 0.937·41-s − 0.152·43-s + 0.875·47-s − 3/7·49-s + 1.64·53-s − 0.512·61-s + 0.248·65-s + 0.488·67-s − 1.17·73-s − 0.900·79-s + 0.650·85-s − 0.635·89-s − 0.419·91-s − 0.820·95-s + 1.42·97-s + 0.0995·101-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

\(L(1)\) \(\approx\) \(2.543855233\)
\(L(\frac12)\) \(\approx\) \(2.543855233\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.09612493936237, −14.58252694137720, −14.06050346072008, −13.37503935119456, −13.03286862798209, −12.57106496528278, −11.97240580892019, −11.39209216046349, −10.63616461398223, −10.25991555809503, −9.846062737537062, −9.120620830473981, −8.573493782198921, −8.185751063448940, −7.276580055452654, −6.791011284801255, −6.094047395002959, −5.874037076494148, −4.941541667291374, −4.380768707443090, −3.585651935712347, −2.959531084035998, −2.377799801488345, −1.355003125574179, −0.6522814720379540, 0.6522814720379540, 1.355003125574179, 2.377799801488345, 2.959531084035998, 3.585651935712347, 4.380768707443090, 4.941541667291374, 5.874037076494148, 6.094047395002959, 6.791011284801255, 7.276580055452654, 8.185751063448940, 8.573493782198921, 9.120620830473981, 9.846062737537062, 10.25991555809503, 10.63616461398223, 11.39209216046349, 11.97240580892019, 12.57106496528278, 13.03286862798209, 13.37503935119456, 14.06050346072008, 14.58252694137720, 15.09612493936237

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