Properties

Label 2-7360-1.1-c1-0-11
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  − 0.352·3-s − 5-s − 3.95·7-s − 2.87·9-s + 5.63·11-s − 2.53·13-s + 0.352·15-s + 1.26·17-s − 6.83·19-s + 1.39·21-s + 23-s + 25-s + 2.07·27-s − 7.69·29-s − 9.04·31-s − 1.98·33-s + 3.95·35-s + 2.90·37-s + 0.893·39-s + 0.188·41-s − 2.97·43-s + 2.87·45-s − 4.82·47-s + 8.61·49-s − 0.445·51-s − 0.328·53-s − 5.63·55-s + ⋯
L(s)  = 1  − 0.203·3-s − 0.447·5-s − 1.49·7-s − 0.958·9-s + 1.70·11-s − 0.702·13-s + 0.0909·15-s + 0.306·17-s − 1.56·19-s + 0.303·21-s + 0.208·23-s + 0.200·25-s + 0.398·27-s − 1.42·29-s − 1.62·31-s − 0.345·33-s + 0.667·35-s + 0.477·37-s + 0.143·39-s + 0.0294·41-s − 0.453·43-s + 0.428·45-s − 0.704·47-s + 1.23·49-s − 0.0623·51-s − 0.0451·53-s − 0.760·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5511743569\)
\(L(\frac12)\) \(\approx\) \(0.5511743569\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.352T + 3T^{2} \)
7 \( 1 + 3.95T + 7T^{2} \)
11 \( 1 - 5.63T + 11T^{2} \)
13 \( 1 + 2.53T + 13T^{2} \)
17 \( 1 - 1.26T + 17T^{2} \)
19 \( 1 + 6.83T + 19T^{2} \)
29 \( 1 + 7.69T + 29T^{2} \)
31 \( 1 + 9.04T + 31T^{2} \)
37 \( 1 - 2.90T + 37T^{2} \)
41 \( 1 - 0.188T + 41T^{2} \)
43 \( 1 + 2.97T + 43T^{2} \)
47 \( 1 + 4.82T + 47T^{2} \)
53 \( 1 + 0.328T + 53T^{2} \)
59 \( 1 - 0.966T + 59T^{2} \)
61 \( 1 + 1.31T + 61T^{2} \)
67 \( 1 + 3.94T + 67T^{2} \)
71 \( 1 - 7.69T + 71T^{2} \)
73 \( 1 - 0.617T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 - 1.30T + 83T^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 - 8.26T + 97T^{2} \)
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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

−7.88287781304940750682889070561, −6.97884041055053617813815704068, −6.55310797327961204029413490081, −5.96923069532221371582992838176, −5.18953519689984279376644320506, −4.05153229639193568121532973050, −3.66445610968067601448711441890, −2.84998095056884502270549978898, −1.81768627731064273632532663897, −0.36051147633340997923630196605, 0.36051147633340997923630196605, 1.81768627731064273632532663897, 2.84998095056884502270549978898, 3.66445610968067601448711441890, 4.05153229639193568121532973050, 5.18953519689984279376644320506, 5.96923069532221371582992838176, 6.55310797327961204029413490081, 6.97884041055053617813815704068, 7.88287781304940750682889070561

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