Properties

Label 2-7605-1.1-c1-0-65
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  − 1.21·2-s − 0.512·4-s + 5-s + 3.60·7-s + 3.06·8-s − 1.21·10-s − 5.37·11-s − 4.39·14-s − 2.71·16-s + 1.13·17-s + 2.26·19-s − 0.512·20-s + 6.55·22-s + 3.89·23-s + 25-s − 1.84·28-s + 0.0247·29-s + 5.46·31-s − 2.81·32-s − 1.38·34-s + 3.60·35-s − 8.70·37-s − 2.76·38-s + 3.06·40-s − 3.73·41-s − 1.13·43-s + 2.75·44-s + ⋯
L(s)  = 1  − 0.862·2-s − 0.256·4-s + 0.447·5-s + 1.36·7-s + 1.08·8-s − 0.385·10-s − 1.61·11-s − 1.17·14-s − 0.678·16-s + 0.274·17-s + 0.520·19-s − 0.114·20-s + 1.39·22-s + 0.811·23-s + 0.200·25-s − 0.348·28-s + 0.00459·29-s + 0.981·31-s − 0.498·32-s − 0.236·34-s + 0.608·35-s − 1.43·37-s − 0.448·38-s + 0.484·40-s − 0.582·41-s − 0.172·43-s + 0.414·44-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

\(L(1)\) \(\approx\) \(1.310935876\)
\(L(\frac12)\) \(\approx\) \(1.310935876\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.21T + 2T^{2} \)
7 \( 1 - 3.60T + 7T^{2} \)
11 \( 1 + 5.37T + 11T^{2} \)
17 \( 1 - 1.13T + 17T^{2} \)
19 \( 1 - 2.26T + 19T^{2} \)
23 \( 1 - 3.89T + 23T^{2} \)
29 \( 1 - 0.0247T + 29T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + 8.70T + 37T^{2} \)
41 \( 1 + 3.73T + 41T^{2} \)
43 \( 1 + 1.13T + 43T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 - 0.171T + 59T^{2} \)
61 \( 1 - 3.36T + 61T^{2} \)
67 \( 1 - 6.39T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 4.70T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−7.996147716252474014066691035762, −7.50332924317211506253640937764, −6.74346071028218048673009468258, −5.49334672492657388579433186380, −5.09134396865681225531407185640, −4.63857679301806920129171115337, −3.44007545219917676071486130728, −2.39736940936600332719783038790, −1.62016166265135182658613918251, −0.68759074507928419909804053108, 0.68759074507928419909804053108, 1.62016166265135182658613918251, 2.39736940936600332719783038790, 3.44007545219917676071486130728, 4.63857679301806920129171115337, 5.09134396865681225531407185640, 5.49334672492657388579433186380, 6.74346071028218048673009468258, 7.50332924317211506253640937764, 7.996147716252474014066691035762

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