Properties

Label 2-1960-1.1-c1-0-9
Degree $2$
Conductor $1960$
Sign $1$
Analytic cond. $15.6506$
Root an. cond. $3.95609$
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 + 4·23-s + 25-s − 2·29-s + 8·31-s + 6·37-s + 6·41-s − 8·43-s + 3·45-s − 4·47-s + 6·53-s − 4·55-s + 4·59-s + 2·61-s − 2·65-s + 8·67-s + 6·73-s + 9·81-s + 16·83-s + 2·85-s + 6·89-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 + 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s − 1.21·43-s + 0.447·45-s − 0.583·47-s + 0.824·53-s − 0.539·55-s + 0.520·59-s + 0.256·61-s − 0.248·65-s + 0.977·67-s + 0.702·73-s + 81-s + 1.75·83-s + 0.216·85-s + 0.635·89-s + 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.526149348\)
\(L(\frac12)\) \(\approx\) \(1.526149348\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−8.938571023394552259974771089914, −8.568492662536040706055181654407, −7.73148938409257749345471205858, −6.56407051249131463125622663292, −6.27788039497575849944468549246, −5.08516938187217910881281818728, −4.17940013953949332311038958729, −3.38301616382596410288851087260, −2.29215136886722219541328805003, −0.834108506722948485247590602076, 0.834108506722948485247590602076, 2.29215136886722219541328805003, 3.38301616382596410288851087260, 4.17940013953949332311038958729, 5.08516938187217910881281818728, 6.27788039497575849944468549246, 6.56407051249131463125622663292, 7.73148938409257749345471205858, 8.568492662536040706055181654407, 8.938571023394552259974771089914

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