Properties

Label 2-7605-1.1-c1-0-60
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  − 2.49·2-s + 4.22·4-s + 5-s + 1.90·7-s − 5.55·8-s − 2.49·10-s − 1.06·11-s − 4.75·14-s + 5.41·16-s − 0.637·17-s + 5.73·19-s + 4.22·20-s + 2.66·22-s − 3.81·23-s + 25-s + 8.05·28-s − 9.45·29-s − 1.46·31-s − 2.40·32-s + 1.59·34-s + 1.90·35-s + 0.757·37-s − 14.3·38-s − 5.55·40-s − 0.267·41-s + 0.637·43-s − 4.52·44-s + ⋯
L(s)  = 1  − 1.76·2-s + 2.11·4-s + 0.447·5-s + 0.720·7-s − 1.96·8-s − 0.789·10-s − 0.322·11-s − 1.27·14-s + 1.35·16-s − 0.154·17-s + 1.31·19-s + 0.945·20-s + 0.568·22-s − 0.796·23-s + 0.200·25-s + 1.52·28-s − 1.75·29-s − 0.262·31-s − 0.424·32-s + 0.272·34-s + 0.322·35-s + 0.124·37-s − 2.32·38-s − 0.878·40-s − 0.0418·41-s + 0.0971·43-s − 0.681·44-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

\(L(1)\) \(\approx\) \(0.9298105317\)
\(L(\frac12)\) \(\approx\) \(0.9298105317\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.49T + 2T^{2} \)
7 \( 1 - 1.90T + 7T^{2} \)
11 \( 1 + 1.06T + 11T^{2} \)
17 \( 1 + 0.637T + 17T^{2} \)
19 \( 1 - 5.73T + 19T^{2} \)
23 \( 1 + 3.81T + 23T^{2} \)
29 \( 1 + 9.45T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 - 0.757T + 37T^{2} \)
41 \( 1 + 0.267T + 41T^{2} \)
43 \( 1 - 0.637T + 43T^{2} \)
47 \( 1 - 9.44T + 47T^{2} \)
53 \( 1 - 6.99T + 53T^{2} \)
59 \( 1 + 0.741T + 59T^{2} \)
61 \( 1 - 4.19T + 61T^{2} \)
67 \( 1 - 8.09T + 67T^{2} \)
71 \( 1 + 9.76T + 71T^{2} \)
73 \( 1 + 3.71T + 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 + 5.11T + 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + 4.22T + 97T^{2} \)
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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

−7.87145924951049307576173249375, −7.47202840920975035726322141271, −6.88022226098519582166630404233, −5.84403532152736117614246999415, −5.42379855949288328868716993606, −4.29065238667945428273172559728, −3.18070481855639347135578920699, −2.19953957514865454891235764994, −1.64732387994604418231175904511, −0.64218813108593808816518722240, 0.64218813108593808816518722240, 1.64732387994604418231175904511, 2.19953957514865454891235764994, 3.18070481855639347135578920699, 4.29065238667945428273172559728, 5.42379855949288328868716993606, 5.84403532152736117614246999415, 6.88022226098519582166630404233, 7.47202840920975035726322141271, 7.87145924951049307576173249375

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