Properties

Label 2-7605-1.1-c1-0-96
Degree $2$
Conductor $7605$
Sign $1$
Analytic cond. $60.7262$
Root an. cond. $7.79270$
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  + 0.219·2-s − 1.95·4-s + 5-s − 0.332·7-s − 0.868·8-s + 0.219·10-s + 5.37·11-s − 0.0729·14-s + 3.71·16-s + 5.06·17-s + 2.26·19-s − 1.95·20-s + 1.18·22-s + 2.83·23-s + 25-s + 0.648·28-s + 2.90·29-s + 5.46·31-s + 2.55·32-s + 1.11·34-s − 0.332·35-s + 5.97·37-s + 0.498·38-s − 0.868·40-s − 3.73·41-s − 5.06·43-s − 10.4·44-s + ⋯
L(s)  = 1  + 0.155·2-s − 0.975·4-s + 0.447·5-s − 0.125·7-s − 0.306·8-s + 0.0694·10-s + 1.61·11-s − 0.0195·14-s + 0.928·16-s + 1.22·17-s + 0.520·19-s − 0.436·20-s + 0.251·22-s + 0.592·23-s + 0.200·25-s + 0.122·28-s + 0.539·29-s + 0.981·31-s + 0.451·32-s + 0.190·34-s − 0.0561·35-s + 0.981·37-s + 0.0808·38-s − 0.137·40-s − 0.582·41-s − 0.772·43-s − 1.58·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7605\)    =    \(3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(60.7262\)
Root analytic conductor: \(7.79270\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7605} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7605,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.353599177\)
\(L(\frac12)\) \(\approx\) \(2.353599177\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - 0.219T + 2T^{2} \)
7 \( 1 + 0.332T + 7T^{2} \)
11 \( 1 - 5.37T + 11T^{2} \)
17 \( 1 - 5.06T + 17T^{2} \)
19 \( 1 - 2.26T + 19T^{2} \)
23 \( 1 - 2.83T + 23T^{2} \)
29 \( 1 - 2.90T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 - 5.97T + 37T^{2} \)
41 \( 1 + 3.73T + 41T^{2} \)
43 \( 1 + 5.06T + 43T^{2} \)
47 \( 1 + 8.34T + 47T^{2} \)
53 \( 1 - 1.56T + 53T^{2} \)
59 \( 1 + 2.70T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 9.68T + 73T^{2} \)
79 \( 1 - 4.51T + 79T^{2} \)
83 \( 1 - 4.26T + 83T^{2} \)
89 \( 1 + 3.22T + 89T^{2} \)
97 \( 1 + 2.50T + 97T^{2} \)
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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

−8.105786344488188113508304512051, −7.04939199783332093064007320743, −6.43278012809285356842520443278, −5.74475316094223701652324766066, −5.02181359694962127006887578588, −4.36525307011471871736990959294, −3.53327570218927946350942303765, −2.97423489740894986468448335840, −1.51316113151363091522723459848, −0.847599902754695506243994549040, 0.847599902754695506243994549040, 1.51316113151363091522723459848, 2.97423489740894986468448335840, 3.53327570218927946350942303765, 4.36525307011471871736990959294, 5.02181359694962127006887578588, 5.74475316094223701652324766066, 6.43278012809285356842520443278, 7.04939199783332093064007320743, 8.105786344488188113508304512051

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