Properties

Label 2-370-1.1-c1-0-1
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 − 3.34·3-s + 4-s − 5-s − 3.34·6-s − 2.59·7-s + 8-s + 8.19·9-s − 10-s + 4.74·11-s − 3.34·12-s + 6.69·13-s − 2.59·14-s + 3.34·15-s + 16-s − 0.748·17-s + 8.19·18-s − 3.34·19-s − 20-s + 8.69·21-s + 4.74·22-s + 1.49·23-s − 3.34·24-s + 25-s + 6.69·26-s − 17.3·27-s − 2.59·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.93·3-s + 0.5·4-s − 0.447·5-s − 1.36·6-s − 0.981·7-s + 0.353·8-s + 2.73·9-s − 0.316·10-s + 1.43·11-s − 0.965·12-s + 1.85·13-s − 0.694·14-s + 0.863·15-s + 0.250·16-s − 0.181·17-s + 1.93·18-s − 0.767·19-s − 0.223·20-s + 1.89·21-s + 1.01·22-s + 0.312·23-s − 0.682·24-s + 0.200·25-s + 1.31·26-s − 3.34·27-s − 0.490·28-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

\(L(1)\) \(\approx\) \(1.087519120\)
\(L(\frac12)\) \(\approx\) \(1.087519120\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.34T + 3T^{2} \)
7 \( 1 + 2.59T + 7T^{2} \)
11 \( 1 - 4.74T + 11T^{2} \)
13 \( 1 - 6.69T + 13T^{2} \)
17 \( 1 + 0.748T + 17T^{2} \)
19 \( 1 + 3.34T + 19T^{2} \)
23 \( 1 - 1.49T + 23T^{2} \)
29 \( 1 - 3.94T + 29T^{2} \)
31 \( 1 - 7.79T + 31T^{2} \)
41 \( 1 + 6.44T + 41T^{2} \)
43 \( 1 - 1.94T + 43T^{2} \)
47 \( 1 - 1.84T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 - 5.84T + 59T^{2} \)
61 \( 1 - 7.94T + 61T^{2} \)
67 \( 1 - 1.84T + 67T^{2} \)
71 \( 1 + 3.88T + 71T^{2} \)
73 \( 1 + 7.49T + 73T^{2} \)
79 \( 1 + 16.5T + 79T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−11.60180156604721476862487246901, −10.81050962507551854439212411753, −10.01293868207377400237488112873, −8.637363163196287707705750464790, −6.81645835521015055774884096073, −6.51753248776819812170403466852, −5.76275040668198468447716813559, −4.41345416412251328155331739465, −3.70727608921289895429051350715, −1.08962700757098009902654475464, 1.08962700757098009902654475464, 3.70727608921289895429051350715, 4.41345416412251328155331739465, 5.76275040668198468447716813559, 6.51753248776819812170403466852, 6.81645835521015055774884096073, 8.637363163196287707705750464790, 10.01293868207377400237488112873, 10.81050962507551854439212411753, 11.60180156604721476862487246901

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