Properties

Label 2-9360-1.1-c1-0-46
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 4·11-s + 13-s − 2·17-s + 4·19-s + 8·23-s + 25-s + 2·29-s + 8·31-s + 6·37-s + 6·41-s + 4·43-s − 8·47-s − 7·49-s − 6·53-s − 4·55-s − 12·59-s − 2·61-s − 65-s + 4·67-s − 6·73-s − 16·79-s − 4·83-s + 2·85-s − 10·89-s − 4·95-s + 18·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.20·11-s + 0.277·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s − 49-s − 0.824·53-s − 0.539·55-s − 1.56·59-s − 0.256·61-s − 0.124·65-s + 0.488·67-s − 0.702·73-s − 1.80·79-s − 0.439·83-s + 0.216·85-s − 1.05·89-s − 0.410·95-s + 1.82·97-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: \(0\)
Selberg data: \((2,\ 9360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.322592688\)
\(L(\frac12)\) \(\approx\) \(2.322592688\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−7.70207641696339484920839241991, −6.96803123102703029709093160118, −6.43695550060445450001023656796, −5.76579822409874646510496136587, −4.62981604092414037986499809056, −4.44347772636458903616706324481, −3.30362953635638720467719829301, −2.87000851906639316362986982057, −1.52071592036882198484496349519, −0.798842361418802905667250309202, 0.798842361418802905667250309202, 1.52071592036882198484496349519, 2.87000851906639316362986982057, 3.30362953635638720467719829301, 4.44347772636458903616706324481, 4.62981604092414037986499809056, 5.76579822409874646510496136587, 6.43695550060445450001023656796, 6.96803123102703029709093160118, 7.70207641696339484920839241991

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