Properties

Label 2-7360-1.1-c1-0-5
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  − 2.56·3-s − 5-s − 1.56·7-s + 3.56·9-s − 2·11-s − 0.561·13-s + 2.56·15-s − 1.56·17-s − 6·19-s + 4·21-s + 23-s + 25-s − 1.43·27-s + 2.12·29-s − 9.24·31-s + 5.12·33-s + 1.56·35-s + 0.438·37-s + 1.43·39-s − 4.12·41-s − 3.56·45-s − 7.68·47-s − 4.56·49-s + 4·51-s + 0.438·53-s + 2·55-s + 15.3·57-s + ⋯
L(s)  = 1  − 1.47·3-s − 0.447·5-s − 0.590·7-s + 1.18·9-s − 0.603·11-s − 0.155·13-s + 0.661·15-s − 0.378·17-s − 1.37·19-s + 0.872·21-s + 0.208·23-s + 0.200·25-s − 0.276·27-s + 0.394·29-s − 1.66·31-s + 0.891·33-s + 0.263·35-s + 0.0720·37-s + 0.230·39-s − 0.643·41-s − 0.530·45-s − 1.12·47-s − 0.651·49-s + 0.560·51-s + 0.0602·53-s + 0.269·55-s + 2.03·57-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

\(L(1)\) \(\approx\) \(0.1289967899\)
\(L(\frac12)\) \(\approx\) \(0.1289967899\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.56T + 3T^{2} \)
7 \( 1 + 1.56T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 0.561T + 13T^{2} \)
17 \( 1 + 1.56T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
29 \( 1 - 2.12T + 29T^{2} \)
31 \( 1 + 9.24T + 31T^{2} \)
37 \( 1 - 0.438T + 37T^{2} \)
41 \( 1 + 4.12T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 7.68T + 47T^{2} \)
53 \( 1 - 0.438T + 53T^{2} \)
59 \( 1 + 8.68T + 59T^{2} \)
61 \( 1 + 1.12T + 61T^{2} \)
67 \( 1 - 4.43T + 67T^{2} \)
71 \( 1 - 1.87T + 71T^{2} \)
73 \( 1 + 8.56T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 + 2.24T + 89T^{2} \)
97 \( 1 + 4.87T + 97T^{2} \)
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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

−7.80337201638930396026904987793, −6.90705818757530641793191366558, −6.55186184829324675613080525884, −5.82523467346732724361261848658, −5.12464647385414978552669869920, −4.53166996611470291734280220608, −3.70876848149490520475873350230, −2.72982507247659226570294411801, −1.59020388430224735192338928215, −0.19297313723472051299983867567, 0.19297313723472051299983867567, 1.59020388430224735192338928215, 2.72982507247659226570294411801, 3.70876848149490520475873350230, 4.53166996611470291734280220608, 5.12464647385414978552669869920, 5.82523467346732724361261848658, 6.55186184829324675613080525884, 6.90705818757530641793191366558, 7.80337201638930396026904987793

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