Properties

Label 2-370-1.1-c1-0-7
Degree $2$
Conductor $370$
Sign $1$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.93·3-s + 4-s − 5-s + 2.93·6-s − 1.31·7-s + 8-s + 5.63·9-s − 10-s − 0.258·11-s + 2.93·12-s − 5.87·13-s − 1.31·14-s − 2.93·15-s + 16-s + 4.25·17-s + 5.63·18-s + 2.93·19-s − 20-s − 3.87·21-s − 0.258·22-s − 8.51·23-s + 2.93·24-s + 25-s − 5.87·26-s + 7.75·27-s − 1.31·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.69·3-s + 0.5·4-s − 0.447·5-s + 1.19·6-s − 0.498·7-s + 0.353·8-s + 1.87·9-s − 0.316·10-s − 0.0780·11-s + 0.848·12-s − 1.63·13-s − 0.352·14-s − 0.758·15-s + 0.250·16-s + 1.03·17-s + 1.32·18-s + 0.674·19-s − 0.223·20-s − 0.846·21-s − 0.0551·22-s − 1.77·23-s + 0.599·24-s + 0.200·25-s − 1.15·26-s + 1.49·27-s − 0.249·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.927119715\)
\(L(\frac12)\) \(\approx\) \(2.927119715\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
37 \( 1 + T \)
good3 \( 1 - 2.93T + 3T^{2} \)
7 \( 1 + 1.31T + 7T^{2} \)
11 \( 1 + 0.258T + 11T^{2} \)
13 \( 1 + 5.87T + 13T^{2} \)
17 \( 1 - 4.25T + 17T^{2} \)
19 \( 1 - 2.93T + 19T^{2} \)
23 \( 1 + 8.51T + 23T^{2} \)
29 \( 1 + 3.61T + 29T^{2} \)
31 \( 1 - 3.95T + 31T^{2} \)
41 \( 1 + 8.89T + 41T^{2} \)
43 \( 1 + 5.61T + 43T^{2} \)
47 \( 1 - 5.57T + 47T^{2} \)
53 \( 1 - 12.8T + 53T^{2} \)
59 \( 1 - 9.57T + 59T^{2} \)
61 \( 1 - 0.380T + 61T^{2} \)
67 \( 1 - 5.57T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 2.51T + 73T^{2} \)
79 \( 1 + 7.69T + 79T^{2} \)
83 \( 1 + 6.17T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−11.82636435869868264500840212163, −9.996826795763793106699149854426, −9.815362902437729114919347854157, −8.404430912473195544754393272519, −7.67374705854895532338206160241, −6.93467493168068283699568623376, −5.32531411687525139171609671547, −4.02831704671153362193075685414, −3.20876494903941789673814290933, −2.19114773505198629580624255052, 2.19114773505198629580624255052, 3.20876494903941789673814290933, 4.02831704671153362193075685414, 5.32531411687525139171609671547, 6.93467493168068283699568623376, 7.67374705854895532338206160241, 8.404430912473195544754393272519, 9.815362902437729114919347854157, 9.996826795763793106699149854426, 11.82636435869868264500840212163

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