Properties

Label 2-7400-1.1-c1-0-135
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  − 2·3-s + 4·7-s + 9-s + 4·11-s − 2·13-s + 2·17-s − 4·19-s − 8·21-s + 4·27-s − 2·31-s − 8·33-s − 37-s + 4·39-s − 10·41-s − 4·43-s − 12·47-s + 9·49-s − 4·51-s − 6·53-s + 8·57-s + 4·59-s − 8·61-s + 4·63-s + 10·67-s − 12·71-s + 16·77-s − 10·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.74·21-s + 0.769·27-s − 0.359·31-s − 1.39·33-s − 0.164·37-s + 0.640·39-s − 1.56·41-s − 0.609·43-s − 1.75·47-s + 9/7·49-s − 0.560·51-s − 0.824·53-s + 1.05·57-s + 0.520·59-s − 1.02·61-s + 0.503·63-s + 1.22·67-s − 1.42·71-s + 1.82·77-s − 1.12·79-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 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−7.44292215816906958912820611826, −6.73775209282965295411997933549, −6.16272705763484348240687909883, −5.33700500928214593960417816748, −4.83631751093228880496015780385, −4.25250507397163018996277333462, −3.22180599466967263920444678392, −1.89292617825160886313275663612, −1.30799571223406957855186917086, 0, 1.30799571223406957855186917086, 1.89292617825160886313275663612, 3.22180599466967263920444678392, 4.25250507397163018996277333462, 4.83631751093228880496015780385, 5.33700500928214593960417816748, 6.16272705763484348240687909883, 6.73775209282965295411997933549, 7.44292215816906958912820611826

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