Properties

Label 2-9360-1.1-c1-0-17
Degree $2$
Conductor $9360$
Sign $1$
Analytic cond. $74.7399$
Root an. cond. $8.64522$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s + 0.449·11-s − 13-s + 2.89·17-s + 4.44·19-s + 1.55·23-s + 25-s − 4·29-s − 0.449·31-s + 2·35-s − 4.89·37-s − 1.10·41-s + 3.34·43-s − 2·47-s − 3·49-s − 10.8·53-s − 0.449·55-s − 5.34·59-s + 13.7·61-s + 65-s + 14.8·67-s − 8.44·71-s + 14.6·73-s − 0.898·77-s + 4.89·79-s + 2·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 0.135·11-s − 0.277·13-s + 0.703·17-s + 1.02·19-s + 0.323·23-s + 0.200·25-s − 0.742·29-s − 0.0807·31-s + 0.338·35-s − 0.805·37-s − 0.171·41-s + 0.510·43-s − 0.291·47-s − 0.428·49-s − 1.49·53-s − 0.0606·55-s − 0.696·59-s + 1.76·61-s + 0.124·65-s + 1.82·67-s − 1.00·71-s + 1.72·73-s − 0.102·77-s + 0.551·79-s + 0.219·83-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.428176118\)
\(L(\frac12)\) \(\approx\) \(1.428176118\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 0.449T + 11T^{2} \)
17 \( 1 - 2.89T + 17T^{2} \)
19 \( 1 - 4.44T + 19T^{2} \)
23 \( 1 - 1.55T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 0.449T + 31T^{2} \)
37 \( 1 + 4.89T + 37T^{2} \)
41 \( 1 + 1.10T + 41T^{2} \)
43 \( 1 - 3.34T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 5.34T + 59T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 + 8.44T + 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
79 \( 1 - 4.89T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−7.77205440815348014317077570762, −6.94662088522018848188914207050, −6.53255652006910285148653264852, −5.50737844115263622899769285868, −5.11228072166909072334801041711, −4.04653808663558234924783036951, −3.42448266190533901712907782464, −2.82630205624995873760665583718, −1.66697266065152779182307771343, −0.57650592475608153199575526950, 0.57650592475608153199575526950, 1.66697266065152779182307771343, 2.82630205624995873760665583718, 3.42448266190533901712907782464, 4.04653808663558234924783036951, 5.11228072166909072334801041711, 5.50737844115263622899769285868, 6.53255652006910285148653264852, 6.94662088522018848188914207050, 7.77205440815348014317077570762

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