Properties

Label 2-7400-1.1-c1-0-38
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 + 7-s − 2·9-s − 3·11-s + 8·17-s − 21-s + 4·23-s + 5·27-s − 8·29-s + 6·31-s + 3·33-s − 37-s + 3·41-s − 2·43-s − 3·47-s − 6·49-s − 8·51-s + 53-s − 4·59-s − 6·61-s − 2·63-s − 4·67-s − 4·69-s − 71-s − 9·73-s − 3·77-s + 10·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 1.94·17-s − 0.218·21-s + 0.834·23-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 − 0.304·43-s − 0.437·47-s − 6/7·49-s − 1.12·51-s + 0.137·53-s − 0.520·59-s − 0.768·61-s − 0.251·63-s − 0.488·67-s − 0.481·69-s − 0.118·71-s − 1.05·73-s − 0.341·77-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: \(0\)
Selberg data: \((2,\ 7400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.353558826\)
\(L(\frac12)\) \(\approx\) \(1.353558826\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.84398251714134607202379019235, −7.35377320011858851608234839219, −6.35065534761932545422422226113, −5.67293785664974242218224268682, −5.23634755551812806796461416296, −4.56256051694647410680413706936, −3.35711251950245347198613066120, −2.87109977733041583294426088815, −1.67165218971909948158549267416, −0.61064235119415058309456258343, 0.61064235119415058309456258343, 1.67165218971909948158549267416, 2.87109977733041583294426088815, 3.35711251950245347198613066120, 4.56256051694647410680413706936, 5.23634755551812806796461416296, 5.67293785664974242218224268682, 6.35065534761932545422422226113, 7.35377320011858851608234839219, 7.84398251714134607202379019235

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