Properties

Label 2-2960-1.1-c1-0-17
Degree $2$
Conductor $2960$
Sign $1$
Analytic cond. $23.6357$
Root an. cond. $4.86165$
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 − 3·9-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s + 25-s − 6·29-s + 4·31-s − 37-s − 6·41-s − 4·43-s + 3·45-s + 8·47-s − 7·49-s + 10·53-s − 4·55-s − 4·59-s + 10·61-s − 2·65-s + 8·67-s + 10·73-s + 4·79-s + 9·81-s + 2·85-s + 2·89-s − 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.164·37-s − 0.937·41-s − 0.609·43-s + 0.447·45-s + 1.16·47-s − 49-s + 1.37·53-s − 0.539·55-s − 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.977·67-s + 1.17·73-s + 0.450·79-s + 81-s + 0.216·85-s + 0.211·89-s − 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(23.6357\)
Root analytic conductor: \(4.86165\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.602782834\)
\(L(\frac12)\) \(\approx\) \(1.602782834\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
37 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−8.734531122003643087724049825453, −8.129825714537106723073907080795, −7.18886119243338118366578821365, −6.49664616322372714025027522541, −5.73020676976688720839306571574, −4.88856578354223850475874803229, −3.82494672208517844438568110208, −3.30347465535627875246494353467, −2.06558083903480564080310734566, −0.78024997926188957232556441579, 0.78024997926188957232556441579, 2.06558083903480564080310734566, 3.30347465535627875246494353467, 3.82494672208517844438568110208, 4.88856578354223850475874803229, 5.73020676976688720839306571574, 6.49664616322372714025027522541, 7.18886119243338118366578821365, 8.129825714537106723073907080795, 8.734531122003643087724049825453

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