Properties

Label 2-30960-1.1-c1-0-55
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 6·13-s − 2·17-s − 2·19-s − 6·23-s + 25-s − 6·29-s − 4·31-s − 8·37-s − 8·41-s − 43-s − 6·47-s − 7·49-s − 6·53-s − 4·59-s − 14·61-s + 6·65-s + 4·67-s + 8·71-s − 4·73-s + 12·79-s + 2·83-s + 2·85-s + 14·89-s + 2·95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.66·13-s − 0.485·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 1.31·37-s − 1.24·41-s − 0.152·43-s − 0.875·47-s − 49-s − 0.824·53-s − 0.520·59-s − 1.79·61-s + 0.744·65-s + 0.488·67-s + 0.949·71-s − 0.468·73-s + 1.35·79-s + 0.219·83-s + 0.216·85-s + 1.48·89-s + 0.205·95-s − 0.203·97-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: \(2\)
Selberg data: \((2,\ 30960,\ (\ :1/2),\ 1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
43 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.54872909389621, −15.08250775886423, −14.61864422567251, −14.15543698451094, −13.50071795787453, −12.92990346177129, −12.31600942827714, −12.04245506204743, −11.43650711336482, −10.77279815624374, −10.34117093177609, −9.606757418974514, −9.291048100601787, −8.538710640063080, −7.797901428708296, −7.644210830210262, −6.714292760863829, −6.463794931790917, −5.420178627488385, −5.010172241837863, −4.374285942599860, −3.664280982197907, −3.060312321667707, −2.096430531278342, −1.704893012055578, 0, 0, 1.704893012055578, 2.096430531278342, 3.060312321667707, 3.664280982197907, 4.374285942599860, 5.010172241837863, 5.420178627488385, 6.463794931790917, 6.714292760863829, 7.644210830210262, 7.797901428708296, 8.538710640063080, 9.291048100601787, 9.606757418974514, 10.34117093177609, 10.77279815624374, 11.43650711336482, 12.04245506204743, 12.31600942827714, 12.92990346177129, 13.50071795787453, 14.15543698451094, 14.61864422567251, 15.08250775886423, 15.54872909389621

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