Properties

Label 2-7360-1.1-c1-0-136
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  + 1.80·3-s + 5-s + 4.65·7-s + 0.265·9-s + 4.38·11-s + 4.79·13-s + 1.80·15-s + 0.285·17-s + 3.30·19-s + 8.40·21-s − 23-s + 25-s − 4.94·27-s + 3.47·29-s − 2.44·31-s + 7.91·33-s + 4.65·35-s − 11.0·37-s + 8.66·39-s + 11.7·41-s − 2.24·43-s + 0.265·45-s − 8.28·47-s + 14.6·49-s + 0.515·51-s − 3.81·53-s + 4.38·55-s + ⋯
L(s)  = 1  + 1.04·3-s + 0.447·5-s + 1.75·7-s + 0.0885·9-s + 1.32·11-s + 1.32·13-s + 0.466·15-s + 0.0691·17-s + 0.759·19-s + 1.83·21-s − 0.208·23-s + 0.200·25-s − 0.950·27-s + 0.644·29-s − 0.439·31-s + 1.37·33-s + 0.786·35-s − 1.81·37-s + 1.38·39-s + 1.83·41-s − 0.343·43-s + 0.0395·45-s − 1.20·47-s + 2.09·49-s + 0.0721·51-s − 0.524·53-s + 0.590·55-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

\(L(1)\) \(\approx\) \(4.964243458\)
\(L(\frac12)\) \(\approx\) \(4.964243458\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.80T + 3T^{2} \)
7 \( 1 - 4.65T + 7T^{2} \)
11 \( 1 - 4.38T + 11T^{2} \)
13 \( 1 - 4.79T + 13T^{2} \)
17 \( 1 - 0.285T + 17T^{2} \)
19 \( 1 - 3.30T + 19T^{2} \)
29 \( 1 - 3.47T + 29T^{2} \)
31 \( 1 + 2.44T + 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 + 2.24T + 43T^{2} \)
47 \( 1 + 8.28T + 47T^{2} \)
53 \( 1 + 3.81T + 53T^{2} \)
59 \( 1 - 9.43T + 59T^{2} \)
61 \( 1 - 5.54T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + 9.87T + 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 - 8.15T + 83T^{2} \)
89 \( 1 - 1.98T + 89T^{2} \)
97 \( 1 - 4.33T + 97T^{2} \)
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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

−8.066250396002627896743677832823, −7.41431651848770364248179944190, −6.55398314145448694896630655519, −5.75751513496079383713513703156, −5.08488356639519425763404217113, −4.14186803774509417457090762311, −3.61140383426831443260231224735, −2.65760796394772530480496964985, −1.61093390561504521606933920247, −1.31657234338124411785084788110, 1.31657234338124411785084788110, 1.61093390561504521606933920247, 2.65760796394772530480496964985, 3.61140383426831443260231224735, 4.14186803774509417457090762311, 5.08488356639519425763404217113, 5.75751513496079383713513703156, 6.55398314145448694896630655519, 7.41431651848770364248179944190, 8.066250396002627896743677832823

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