Properties

Label 2-7360-1.1-c1-0-120
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  − 1.79·3-s + 5-s + 2.79·7-s + 0.208·9-s − 3.79·11-s − 1.20·13-s − 1.79·15-s − 3.79·17-s − 1.20·19-s − 5·21-s + 23-s + 25-s + 5.00·27-s + 1.58·29-s + 10.3·31-s + 6.79·33-s + 2.79·35-s + 4·37-s + 2.16·39-s − 2.20·41-s + 7.16·43-s + 0.208·45-s − 13.5·47-s + 0.791·49-s + 6.79·51-s − 6·53-s − 3.79·55-s + ⋯
L(s)  = 1  − 1.03·3-s + 0.447·5-s + 1.05·7-s + 0.0695·9-s − 1.14·11-s − 0.335·13-s − 0.462·15-s − 0.919·17-s − 0.277·19-s − 1.09·21-s + 0.208·23-s + 0.200·25-s + 0.962·27-s + 0.293·29-s + 1.86·31-s + 1.18·33-s + 0.471·35-s + 0.657·37-s + 0.346·39-s − 0.344·41-s + 1.09·43-s + 0.0311·45-s − 1.98·47-s + 0.113·49-s + 0.950·51-s − 0.824·53-s − 0.511·55-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 \( 1 + 1.79T + 3T^{2} \)
7 \( 1 - 2.79T + 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 + 1.20T + 13T^{2} \)
17 \( 1 + 3.79T + 17T^{2} \)
19 \( 1 + 1.20T + 19T^{2} \)
29 \( 1 - 1.58T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 2.20T + 41T^{2} \)
43 \( 1 - 7.16T + 43T^{2} \)
47 \( 1 + 13.5T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 4.41T + 59T^{2} \)
61 \( 1 - 3.37T + 61T^{2} \)
67 \( 1 - 7.16T + 67T^{2} \)
71 \( 1 + 5.37T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 3.16T + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−7.61526191631514262568548275866, −6.59509008933691159367212668262, −6.21936048580436766920642169472, −5.27311148137433832417669718019, −4.92224953166670623042522099035, −4.34987330605809724263098116840, −2.89714219406380657439587058079, −2.26238286471411208383711708274, −1.15522517300381585841096040172, 0, 1.15522517300381585841096040172, 2.26238286471411208383711708274, 2.89714219406380657439587058079, 4.34987330605809724263098116840, 4.92224953166670623042522099035, 5.27311148137433832417669718019, 6.21936048580436766920642169472, 6.59509008933691159367212668262, 7.61526191631514262568548275866

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