Properties

Label 2-9360-1.1-c1-0-118
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 + 2.71·7-s − 2.71·11-s + 13-s + 2.83·17-s + 3.55·19-s − 4.83·23-s + 25-s − 6·29-s − 7.55·31-s + 2.71·35-s − 4.27·37-s − 2.83·41-s − 11.1·43-s − 11.5·47-s + 0.397·49-s − 1.16·53-s − 2.71·55-s − 2.11·59-s + 6.60·61-s + 65-s − 1.88·67-s − 6.71·71-s + 9.11·73-s − 7.39·77-s − 10.2·79-s + 2.11·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.02·7-s − 0.820·11-s + 0.277·13-s + 0.688·17-s + 0.816·19-s − 1.00·23-s + 0.200·25-s − 1.11·29-s − 1.35·31-s + 0.459·35-s − 0.703·37-s − 0.443·41-s − 1.69·43-s − 1.68·47-s + 0.0567·49-s − 0.159·53-s − 0.366·55-s − 0.275·59-s + 0.845·61-s + 0.124·65-s − 0.230·67-s − 0.797·71-s + 1.06·73-s − 0.843·77-s − 1.15·79-s + 0.232·83-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 \( 1 - 2.71T + 7T^{2} \)
11 \( 1 + 2.71T + 11T^{2} \)
17 \( 1 - 2.83T + 17T^{2} \)
19 \( 1 - 3.55T + 19T^{2} \)
23 \( 1 + 4.83T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 7.55T + 31T^{2} \)
37 \( 1 + 4.27T + 37T^{2} \)
41 \( 1 + 2.83T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 1.16T + 53T^{2} \)
59 \( 1 + 2.11T + 59T^{2} \)
61 \( 1 - 6.60T + 61T^{2} \)
67 \( 1 + 1.88T + 67T^{2} \)
71 \( 1 + 6.71T + 71T^{2} \)
73 \( 1 - 9.11T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 2.11T + 83T^{2} \)
89 \( 1 + 1.16T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−7.45518515716597298897710960843, −6.76529691164989551310313068313, −5.75239271112804467968828778615, −5.35547693496961393329091092477, −4.79271731615888810487528844481, −3.74060216563610477590844572368, −3.09777465896630303930439162194, −1.92914235376511877405223222358, −1.51245668206700871652279007050, 0, 1.51245668206700871652279007050, 1.92914235376511877405223222358, 3.09777465896630303930439162194, 3.74060216563610477590844572368, 4.79271731615888810487528844481, 5.35547693496961393329091092477, 5.75239271112804467968828778615, 6.76529691164989551310313068313, 7.45518515716597298897710960843

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