Properties

Label 2-7605-1.1-c1-0-69
Degree $2$
Conductor $7605$
Sign $1$
Analytic cond. $60.7262$
Root an. cond. $7.79270$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 5-s + 7-s − 3·11-s + 4·16-s + 3·17-s + 4·19-s − 2·20-s + 9·23-s + 25-s − 2·28-s + 6·29-s − 2·31-s + 35-s + 37-s − 3·41-s + 2·43-s + 6·44-s − 6·47-s − 6·49-s − 9·53-s − 3·55-s − 12·59-s + 5·61-s − 8·64-s + 4·67-s − 6·68-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s + 0.377·7-s − 0.904·11-s + 16-s + 0.727·17-s + 0.917·19-s − 0.447·20-s + 1.87·23-s + 1/5·25-s − 0.377·28-s + 1.11·29-s − 0.359·31-s + 0.169·35-s + 0.164·37-s − 0.468·41-s + 0.304·43-s + 0.904·44-s − 0.875·47-s − 6/7·49-s − 1.23·53-s − 0.404·55-s − 1.56·59-s + 0.640·61-s − 64-s + 0.488·67-s − 0.727·68-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7605,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.801221470\)
\(L(\frac12)\) \(\approx\) \(1.801221470\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−7.899852757808751196863738266821, −7.37847609144377513735920143106, −6.41094706066728876391037150030, −5.56895029981771839658716727502, −4.96922748610343118432089441920, −4.66190946366239145964821100773, −3.34184374186132703964531735348, −2.95068554311414819768064013502, −1.59745923283821507583674432057, −0.71817739384042992252886876843, 0.71817739384042992252886876843, 1.59745923283821507583674432057, 2.95068554311414819768064013502, 3.34184374186132703964531735348, 4.66190946366239145964821100773, 4.96922748610343118432089441920, 5.56895029981771839658716727502, 6.41094706066728876391037150030, 7.37847609144377513735920143106, 7.899852757808751196863738266821

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