Properties

Label 2-7360-1.1-c1-0-130
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  − 0.602·3-s − 5-s + 4.26·7-s − 2.63·9-s + 3.63·11-s − 4.55·13-s + 0.602·15-s − 1.85·17-s − 1.23·19-s − 2.57·21-s + 23-s + 25-s + 3.39·27-s − 5.38·29-s + 4.49·31-s − 2.19·33-s − 4.26·35-s + 1.78·37-s + 2.74·39-s − 2.90·41-s + 0.649·43-s + 2.63·45-s + 5.27·47-s + 11.1·49-s + 1.11·51-s − 4.45·53-s − 3.63·55-s + ⋯
L(s)  = 1  − 0.348·3-s − 0.447·5-s + 1.61·7-s − 0.878·9-s + 1.09·11-s − 1.26·13-s + 0.155·15-s − 0.449·17-s − 0.284·19-s − 0.560·21-s + 0.208·23-s + 0.200·25-s + 0.654·27-s − 1.00·29-s + 0.806·31-s − 0.382·33-s − 0.720·35-s + 0.293·37-s + 0.440·39-s − 0.452·41-s + 0.0990·43-s + 0.393·45-s + 0.768·47-s + 1.59·49-s + 0.156·51-s − 0.612·53-s − 0.490·55-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 \( 1 + 0.602T + 3T^{2} \)
7 \( 1 - 4.26T + 7T^{2} \)
11 \( 1 - 3.63T + 11T^{2} \)
13 \( 1 + 4.55T + 13T^{2} \)
17 \( 1 + 1.85T + 17T^{2} \)
19 \( 1 + 1.23T + 19T^{2} \)
29 \( 1 + 5.38T + 29T^{2} \)
31 \( 1 - 4.49T + 31T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 + 2.90T + 41T^{2} \)
43 \( 1 - 0.649T + 43T^{2} \)
47 \( 1 - 5.27T + 47T^{2} \)
53 \( 1 + 4.45T + 53T^{2} \)
59 \( 1 + 4.22T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + 8.69T + 67T^{2} \)
71 \( 1 - 0.430T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 - 1.77T + 83T^{2} \)
89 \( 1 + 9.83T + 89T^{2} \)
97 \( 1 - 1.29T + 97T^{2} \)
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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

−7.64978906579166665330987326647, −6.93120468809714324293661452629, −6.12034135035866312715856389426, −5.35857418609562347101419282456, −4.64044633705339842502267842504, −4.25462923345158982797015768199, −3.09263303944460162851190802764, −2.17686100710496604096870845665, −1.28127974113570154529229658502, 0, 1.28127974113570154529229658502, 2.17686100710496604096870845665, 3.09263303944460162851190802764, 4.25462923345158982797015768199, 4.64044633705339842502267842504, 5.35857418609562347101419282456, 6.12034135035866312715856389426, 6.93120468809714324293661452629, 7.64978906579166665330987326647

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