Properties

Label 2-7605-1.1-c1-0-56
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-s − 4-s + 5-s − 2·7-s − 3·8-s + 10-s + 4·11-s − 2·14-s − 16-s − 4·17-s − 6·19-s − 20-s + 4·22-s + 25-s + 2·28-s − 4·29-s + 10·31-s + 5·32-s − 4·34-s − 2·35-s + 2·37-s − 6·38-s − 3·40-s + 6·41-s − 8·43-s − 4·44-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s − 1.06·8-s + 0.316·10-s + 1.20·11-s − 0.534·14-s − 1/4·16-s − 0.970·17-s − 1.37·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.377·28-s − 0.742·29-s + 1.79·31-s + 0.883·32-s − 0.685·34-s − 0.338·35-s + 0.328·37-s − 0.973·38-s − 0.474·40-s + 0.937·41-s − 1.21·43-s − 0.603·44-s + 1.16·47-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

\(L(1)\) \(\approx\) \(1.953034825\)
\(L(\frac12)\) \(\approx\) \(1.953034825\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.969043383288180295672971025530, −6.72331256200456187347075274964, −6.40963898319760234362375585138, −5.93533235523521642025096572921, −4.88499596390384799429982035157, −4.32037853433109369450804167112, −3.71200381355792790661102239489, −2.85766779133093065341527875930, −1.96176909344580164738176933887, −0.61617982003344263708554497898, 0.61617982003344263708554497898, 1.96176909344580164738176933887, 2.85766779133093065341527875930, 3.71200381355792790661102239489, 4.32037853433109369450804167112, 4.88499596390384799429982035157, 5.93533235523521642025096572921, 6.40963898319760234362375585138, 6.72331256200456187347075274964, 7.969043383288180295672971025530

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