Properties

Label 2-9360-1.1-c1-0-116
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 4.56·7-s − 2.56·11-s − 13-s − 2.56·17-s − 3.12·19-s − 6.56·23-s + 25-s + 1.12·29-s − 6·31-s + 4.56·35-s + 1.68·37-s − 0.561·41-s + 5.12·43-s − 2.87·47-s + 13.8·49-s + 7.68·53-s − 2.56·55-s − 12·59-s + 5.68·61-s − 65-s − 13.3·67-s − 14.5·71-s − 6·73-s − 11.6·77-s + 7.68·79-s − 16.4·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.72·7-s − 0.772·11-s − 0.277·13-s − 0.621·17-s − 0.716·19-s − 1.36·23-s + 0.200·25-s + 0.208·29-s − 1.07·31-s + 0.771·35-s + 0.276·37-s − 0.0876·41-s + 0.781·43-s − 0.419·47-s + 1.97·49-s + 1.05·53-s − 0.345·55-s − 1.56·59-s + 0.727·61-s − 0.124·65-s − 1.63·67-s − 1.72·71-s − 0.702·73-s − 1.33·77-s + 0.864·79-s − 1.81·83-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: \(1\)
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 \( 1 - 4.56T + 7T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
17 \( 1 + 2.56T + 17T^{2} \)
19 \( 1 + 3.12T + 19T^{2} \)
23 \( 1 + 6.56T + 23T^{2} \)
29 \( 1 - 1.12T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 1.68T + 37T^{2} \)
41 \( 1 + 0.561T + 41T^{2} \)
43 \( 1 - 5.12T + 43T^{2} \)
47 \( 1 + 2.87T + 47T^{2} \)
53 \( 1 - 7.68T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 5.68T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 7.68T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 + 1.68T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−7.63745624210096357171202597080, −6.70668270789967633890177593551, −5.79986410952386088974612850144, −5.37785113468469007359117471042, −4.49960807842746904731067276608, −4.16076445854621581259954130651, −2.79710430694435538976692478721, −2.07720068936321360482182710078, −1.48507205216074216520246176703, 0, 1.48507205216074216520246176703, 2.07720068936321360482182710078, 2.79710430694435538976692478721, 4.16076445854621581259954130651, 4.49960807842746904731067276608, 5.37785113468469007359117471042, 5.79986410952386088974612850144, 6.70668270789967633890177593551, 7.63745624210096357171202597080

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