Properties

Label 2-7360-1.1-c1-0-125
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.11·3-s + 5-s + 4.50·7-s + 6.72·9-s − 4.33·11-s + 3.72·13-s + 3.11·15-s + 1.11·17-s − 4.50·19-s + 14.0·21-s − 23-s + 25-s + 11.6·27-s + 8.23·29-s + 1.72·31-s − 13.5·33-s + 4.50·35-s + 0.781·37-s + 11.6·39-s + 3.90·41-s − 8·43-s + 6.72·45-s − 11.4·47-s + 13.3·49-s + 3.49·51-s + 6·53-s − 4.33·55-s + ⋯
L(s)  = 1  + 1.80·3-s + 0.447·5-s + 1.70·7-s + 2.24·9-s − 1.30·11-s + 1.03·13-s + 0.805·15-s + 0.271·17-s − 1.03·19-s + 3.06·21-s − 0.208·23-s + 0.200·25-s + 2.23·27-s + 1.52·29-s + 0.310·31-s − 2.35·33-s + 0.762·35-s + 0.128·37-s + 1.86·39-s + 0.609·41-s − 1.21·43-s + 1.00·45-s − 1.67·47-s + 1.90·49-s + 0.488·51-s + 0.824·53-s − 0.584·55-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: \(0\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.755031156\)
\(L(\frac12)\) \(\approx\) \(5.755031156\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.11T + 3T^{2} \)
7 \( 1 - 4.50T + 7T^{2} \)
11 \( 1 + 4.33T + 11T^{2} \)
13 \( 1 - 3.72T + 13T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 + 4.50T + 19T^{2} \)
29 \( 1 - 8.23T + 29T^{2} \)
31 \( 1 - 1.72T + 31T^{2} \)
37 \( 1 - 0.781T + 37T^{2} \)
41 \( 1 - 3.90T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 2.23T + 59T^{2} \)
61 \( 1 + 3.55T + 61T^{2} \)
67 \( 1 + 2.43T + 67T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 + 9.45T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 2.78T + 83T^{2} \)
89 \( 1 + 7.69T + 89T^{2} \)
97 \( 1 + 0.642T + 97T^{2} \)
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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

−8.089790338239970761055291195944, −7.59125011680840384729518130075, −6.70383054537346097592806818639, −5.74943822753079013863137390721, −4.78355638153218370824749000126, −4.39353069738290849511257649550, −3.36547919831496856033373737317, −2.60441006045526280641676364045, −1.95173966026119337589501102907, −1.25211897208122148185778082900, 1.25211897208122148185778082900, 1.95173966026119337589501102907, 2.60441006045526280641676364045, 3.36547919831496856033373737317, 4.39353069738290849511257649550, 4.78355638153218370824749000126, 5.74943822753079013863137390721, 6.70383054537346097592806818639, 7.59125011680840384729518130075, 8.089790338239970761055291195944

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