Properties

Label 2-7360-1.1-c1-0-101
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  + 3.06·3-s − 5-s + 4.06·7-s + 6.41·9-s + 3.06·11-s − 5.41·13-s − 3.06·15-s + 1.72·17-s − 5.41·19-s + 12.4·21-s + 23-s + 25-s + 10.4·27-s − 7.34·29-s + 8.06·31-s + 9.41·33-s − 4.06·35-s + 0.789·37-s − 16.6·39-s + 11.8·41-s + 4·43-s − 6.41·45-s − 1.44·47-s + 9.55·49-s + 5.27·51-s + 1.48·53-s − 3.06·55-s + ⋯
L(s)  = 1  + 1.77·3-s − 0.447·5-s + 1.53·7-s + 2.13·9-s + 0.925·11-s − 1.50·13-s − 0.792·15-s + 0.417·17-s − 1.24·19-s + 2.72·21-s + 0.208·23-s + 0.200·25-s + 2.01·27-s − 1.36·29-s + 1.44·31-s + 1.63·33-s − 0.687·35-s + 0.129·37-s − 2.66·39-s + 1.85·41-s + 0.609·43-s − 0.956·45-s − 0.210·47-s + 1.36·49-s + 0.739·51-s + 0.204·53-s − 0.413·55-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

\(L(1)\) \(\approx\) \(4.786029052\)
\(L(\frac12)\) \(\approx\) \(4.786029052\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.06T + 3T^{2} \)
7 \( 1 - 4.06T + 7T^{2} \)
11 \( 1 - 3.06T + 11T^{2} \)
13 \( 1 + 5.41T + 13T^{2} \)
17 \( 1 - 1.72T + 17T^{2} \)
19 \( 1 + 5.41T + 19T^{2} \)
29 \( 1 + 7.34T + 29T^{2} \)
31 \( 1 - 8.06T + 31T^{2} \)
37 \( 1 - 0.789T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 1.44T + 47T^{2} \)
53 \( 1 - 1.48T + 53T^{2} \)
59 \( 1 - 9.62T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + 0.514T + 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + 5.20T + 97T^{2} \)
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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

−7.951880438863534862016404790023, −7.43624114896869028226293513631, −6.96898248403881146757499617588, −5.75129753444436849903924326463, −4.58563489838552559514187022662, −4.37561726328273381545100169882, −3.57677614078158981341791483956, −2.49613833969055787595444455903, −2.09745186725326121953888859052, −1.07785166957767567754086829764, 1.07785166957767567754086829764, 2.09745186725326121953888859052, 2.49613833969055787595444455903, 3.57677614078158981341791483956, 4.37561726328273381545100169882, 4.58563489838552559514187022662, 5.75129753444436849903924326463, 6.96898248403881146757499617588, 7.43624114896869028226293513631, 7.951880438863534862016404790023

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