Properties

Label 2-7400-1.1-c1-0-152
Degree $2$
Conductor $7400$
Sign $-1$
Analytic cond. $59.0892$
Root an. cond. $7.68695$
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  + 3-s + 3·7-s − 2·9-s − 3·11-s − 2·17-s − 2·19-s + 3·21-s + 6·23-s − 5·27-s − 2·29-s − 4·31-s − 3·33-s − 37-s + 7·41-s − 4·43-s − 47-s + 2·49-s − 2·51-s − 9·53-s − 2·57-s + 8·59-s − 4·61-s − 6·63-s − 12·67-s + 6·69-s − 5·71-s + 13·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·7-s − 2/3·9-s − 0.904·11-s − 0.485·17-s − 0.458·19-s + 0.654·21-s + 1.25·23-s − 0.962·27-s − 0.371·29-s − 0.718·31-s − 0.522·33-s − 0.164·37-s + 1.09·41-s − 0.609·43-s − 0.145·47-s + 2/7·49-s − 0.280·51-s − 1.23·53-s − 0.264·57-s + 1.04·59-s − 0.512·61-s − 0.755·63-s − 1.46·67-s + 0.722·69-s − 0.593·71-s + 1.52·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7400\)    =    \(2^{3} \cdot 5^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(59.0892\)
Root analytic conductor: \(7.68695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
37 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−7.77376636045018410124481720336, −7.03345130053130937265414209682, −6.09627234848789494459929096072, −5.29050268572775056511533310261, −4.82813618661929233529824780855, −3.92053715002080621915691309215, −2.97750687228979700956343588916, −2.34409610435184206396859378143, −1.46595757811433027282420609553, 0, 1.46595757811433027282420609553, 2.34409610435184206396859378143, 2.97750687228979700956343588916, 3.92053715002080621915691309215, 4.82813618661929233529824780855, 5.29050268572775056511533310261, 6.09627234848789494459929096072, 7.03345130053130937265414209682, 7.77376636045018410124481720336

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