Properties

Label 2-9360-1.1-c1-0-71
Degree $2$
Conductor $9360$
Sign $-1$
Analytic cond. $74.7399$
Root an. cond. $8.64522$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s + 4·11-s + 13-s − 6·17-s − 4·23-s + 25-s + 6·29-s + 8·31-s + 4·35-s − 2·37-s − 10·41-s + 4·43-s + 8·47-s + 9·49-s + 2·53-s − 4·55-s + 4·59-s + 14·61-s − 65-s + 12·67-s − 8·71-s − 10·73-s − 16·77-s − 4·83-s + 6·85-s − 10·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 1.20·11-s + 0.277·13-s − 1.45·17-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.676·35-s − 0.328·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.274·53-s − 0.539·55-s + 0.520·59-s + 1.79·61-s − 0.124·65-s + 1.46·67-s − 0.949·71-s − 1.17·73-s − 1.82·77-s − 0.439·83-s + 0.650·85-s − 1.05·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9360\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(74.7399\)
Root analytic conductor: \(8.64522\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9360,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−7.06678960635814296998452210212, −6.62450608409775362528049787289, −6.32572833577105258537633653670, −5.37240048293860209391939773136, −4.20761685630917735071114015490, −4.01101675736314046012719817397, −3.07371648192294242277775373739, −2.33767432224167873966790820719, −1.06531152841758481934401065682, 0, 1.06531152841758481934401065682, 2.33767432224167873966790820719, 3.07371648192294242277775373739, 4.01101675736314046012719817397, 4.20761685630917735071114015490, 5.37240048293860209391939773136, 6.32572833577105258537633653670, 6.62450608409775362528049787289, 7.06678960635814296998452210212

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