Properties

Label 2-7600-1.1-c1-0-91
Degree $2$
Conductor $7600$
Sign $-1$
Analytic cond. $60.6863$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·7-s + 9-s − 4·11-s − 8·17-s + 19-s − 4·21-s + 6·23-s + 4·27-s + 2·29-s + 8·31-s + 8·33-s − 6·41-s + 10·43-s − 6·47-s − 3·49-s + 16·51-s − 2·57-s + 4·59-s + 6·61-s + 2·63-s + 2·67-s − 12·69-s − 16·71-s + 16·73-s − 8·77-s − 8·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.94·17-s + 0.229·19-s − 0.872·21-s + 1.25·23-s + 0.769·27-s + 0.371·29-s + 1.43·31-s + 1.39·33-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s + 2.24·51-s − 0.264·57-s + 0.520·59-s + 0.768·61-s + 0.251·63-s + 0.244·67-s − 1.44·69-s − 1.89·71-s + 1.87·73-s − 0.911·77-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7600\)    =    \(2^{4} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(60.6863\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.39888915908123975140607404256, −6.72473130671464083767301067069, −6.16264190426228114673450844156, −5.23820061062561135247630641189, −4.90434360423453436998426216383, −4.30375379556345661923304978794, −2.95881542902081907629887419521, −2.26102185465037040268758146201, −1.03951515966706711853515439343, 0, 1.03951515966706711853515439343, 2.26102185465037040268758146201, 2.95881542902081907629887419521, 4.30375379556345661923304978794, 4.90434360423453436998426216383, 5.23820061062561135247630641189, 6.16264190426228114673450844156, 6.72473130671464083767301067069, 7.39888915908123975140607404256

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