Properties

Label 2-7605-1.1-c1-0-175
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 $1$

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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7605,\ (\ :1/2),\ -1)\)

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)$
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

−7.20317648226588894878435850138, −6.73776527565038545381999542512, −6.03896304691145738748897444447, −5.53587662728401401127953027312, −4.33782133692622593527616771527, −4.07590574891540434307633999927, −3.31251905765708186129095668421, −2.69418708577891787481258806412, −1.16615682181981048647370665586, 0, 1.16615682181981048647370665586, 2.69418708577891787481258806412, 3.31251905765708186129095668421, 4.07590574891540434307633999927, 4.33782133692622593527616771527, 5.53587662728401401127953027312, 6.03896304691145738748897444447, 6.73776527565038545381999542512, 7.20317648226588894878435850138

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