Properties

Label 2-7360-1.1-c1-0-118
Degree $2$
Conductor $7360$
Sign $-1$
Analytic cond. $58.7698$
Root an. cond. $7.66615$
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  − 3.30·3-s − 5-s + 2.94·7-s + 7.89·9-s + 0.430·11-s + 2.53·13-s + 3.30·15-s − 1.14·17-s + 1.90·19-s − 9.72·21-s + 23-s + 25-s − 16.1·27-s − 0.895·29-s − 2.05·31-s − 1.42·33-s − 2.94·35-s − 0.715·37-s − 8.35·39-s + 4.25·41-s − 7.79·43-s − 7.89·45-s − 11.7·47-s + 1.68·49-s + 3.78·51-s − 2.36·53-s − 0.430·55-s + ⋯
L(s)  = 1  − 1.90·3-s − 0.447·5-s + 1.11·7-s + 2.63·9-s + 0.129·11-s + 0.702·13-s + 0.852·15-s − 0.278·17-s + 0.436·19-s − 2.12·21-s + 0.208·23-s + 0.200·25-s − 3.10·27-s − 0.166·29-s − 0.369·31-s − 0.247·33-s − 0.498·35-s − 0.117·37-s − 1.33·39-s + 0.664·41-s − 1.18·43-s − 1.17·45-s − 1.70·47-s + 0.240·49-s + 0.529·51-s − 0.325·53-s − 0.0581·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7360\)    =    \(2^{6} \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(58.7698\)
Root analytic conductor: \(7.66615\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 + 3.30T + 3T^{2} \)
7 \( 1 - 2.94T + 7T^{2} \)
11 \( 1 - 0.430T + 11T^{2} \)
13 \( 1 - 2.53T + 13T^{2} \)
17 \( 1 + 1.14T + 17T^{2} \)
19 \( 1 - 1.90T + 19T^{2} \)
29 \( 1 + 0.895T + 29T^{2} \)
31 \( 1 + 2.05T + 31T^{2} \)
37 \( 1 + 0.715T + 37T^{2} \)
41 \( 1 - 4.25T + 41T^{2} \)
43 \( 1 + 7.79T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + 2.36T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 3.84T + 61T^{2} \)
67 \( 1 - 4.46T + 67T^{2} \)
71 \( 1 + 0.983T + 71T^{2} \)
73 \( 1 - 8.60T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + 0.182T + 83T^{2} \)
89 \( 1 - 5.54T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−7.49772363634376247131211054450, −6.66440588042868471977503727321, −6.18578836147183757445777818195, −5.33623018482214246747293257894, −4.87198795336724834372116333551, −4.28042161338051063446829003615, −3.39611051791315822632340578257, −1.77906077872668069970201960670, −1.14707281385838525415063039770, 0, 1.14707281385838525415063039770, 1.77906077872668069970201960670, 3.39611051791315822632340578257, 4.28042161338051063446829003615, 4.87198795336724834372116333551, 5.33623018482214246747293257894, 6.18578836147183757445777818195, 6.66440588042868471977503727321, 7.49772363634376247131211054450

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