Properties

Label 2-7360-1.1-c1-0-123
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  + 0.302·3-s + 5-s − 3.30·7-s − 2.90·9-s − 0.302·11-s − 1.30·13-s + 0.302·15-s + 3.69·17-s + 7.90·19-s − 1.00·21-s + 23-s + 25-s − 1.78·27-s − 1.39·29-s − 0.697·31-s − 0.0916·33-s − 3.30·35-s + 9.21·37-s − 0.394·39-s − 8.30·41-s − 4·43-s − 2.90·45-s − 10.6·47-s + 3.90·49-s + 1.11·51-s − 7.21·53-s − 0.302·55-s + ⋯
L(s)  = 1  + 0.174·3-s + 0.447·5-s − 1.24·7-s − 0.969·9-s − 0.0912·11-s − 0.361·13-s + 0.0781·15-s + 0.896·17-s + 1.81·19-s − 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.344·27-s − 0.258·29-s − 0.125·31-s − 0.0159·33-s − 0.558·35-s + 1.51·37-s − 0.0631·39-s − 1.29·41-s − 0.609·43-s − 0.433·45-s − 1.54·47-s + 0.558·49-s + 0.156·51-s − 0.990·53-s − 0.0408·55-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 \( 1 - 0.302T + 3T^{2} \)
7 \( 1 + 3.30T + 7T^{2} \)
11 \( 1 + 0.302T + 11T^{2} \)
13 \( 1 + 1.30T + 13T^{2} \)
17 \( 1 - 3.69T + 17T^{2} \)
19 \( 1 - 7.90T + 19T^{2} \)
29 \( 1 + 1.39T + 29T^{2} \)
31 \( 1 + 0.697T + 31T^{2} \)
37 \( 1 - 9.21T + 37T^{2} \)
41 \( 1 + 8.30T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + 7.21T + 53T^{2} \)
59 \( 1 + 8.60T + 59T^{2} \)
61 \( 1 - 2.30T + 61T^{2} \)
67 \( 1 - 5.21T + 67T^{2} \)
71 \( 1 + 1.69T + 71T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 3.21T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 18.7T + 97T^{2} \)
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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

−7.72461929978765157566273487860, −6.67935056157105064923058187076, −6.26266052959310795509096411371, −5.38409653934961654966137334031, −5.01387005247890360020623258373, −3.54080698225297808582796089022, −3.21428846901977934002638162414, −2.48152779133608152816852651669, −1.22927341271740845674720379465, 0, 1.22927341271740845674720379465, 2.48152779133608152816852651669, 3.21428846901977934002638162414, 3.54080698225297808582796089022, 5.01387005247890360020623258373, 5.38409653934961654966137334031, 6.26266052959310795509096411371, 6.67935056157105064923058187076, 7.72461929978765157566273487860

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