Properties

Label 2-7400-1.1-c1-0-22
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·7-s − 2·9-s − 3·11-s − 7·17-s − 7·19-s − 2·21-s + 6·23-s − 5·27-s + 8·29-s + 6·31-s − 3·33-s − 37-s − 3·41-s − 4·43-s + 4·47-s − 3·49-s − 7·51-s + 6·53-s − 7·57-s + 8·59-s + 6·61-s + 4·63-s + 3·67-s + 6·69-s + 3·73-s + 6·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s − 2/3·9-s − 0.904·11-s − 1.69·17-s − 1.60·19-s − 0.436·21-s + 1.25·23-s − 0.962·27-s + 1.48·29-s + 1.07·31-s − 0.522·33-s − 0.164·37-s − 0.468·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.980·51-s + 0.824·53-s − 0.927·57-s + 1.04·59-s + 0.768·61-s + 0.503·63-s + 0.366·67-s + 0.722·69-s + 0.351·73-s + 0.683·77-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: \(0\)
Selberg data: \((2,\ 7400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.237559736\)
\(L(\frac12)\) \(\approx\) \(1.237559736\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.159097252436846035064467157771, −7.05605770118786445719714691227, −6.60672474972433212058287024202, −5.94124399712318069252545048708, −4.95145289432935266438290319474, −4.36739249870604579396715759774, −3.35926296564821767026176535446, −2.64349638155056926398099616095, −2.17369263311792471177622097678, −0.50455294985716317813456380066, 0.50455294985716317813456380066, 2.17369263311792471177622097678, 2.64349638155056926398099616095, 3.35926296564821767026176535446, 4.36739249870604579396715759774, 4.95145289432935266438290319474, 5.94124399712318069252545048708, 6.60672474972433212058287024202, 7.05605770118786445719714691227, 8.159097252436846035064467157771

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