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Properties

Label 2-370-185.184-c1-0-3
Degree $2$
Conductor $370$
Sign $0.335 - 0.942i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 0.377i·3-s + 4-s + (1.04 + 1.97i)5-s − 0.377i·6-s + 0.631i·7-s − 8-s + 2.85·9-s + (−1.04 − 1.97i)10-s + 1.24·11-s + 0.377i·12-s − 3.34·13-s − 0.631i·14-s + (−0.746 + 0.395i)15-s + 16-s − 3.10·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.218i·3-s + 0.5·4-s + (0.468 + 0.883i)5-s − 0.154i·6-s + 0.238i·7-s − 0.353·8-s + 0.952·9-s + (−0.331 − 0.624i)10-s + 0.376·11-s + 0.109i·12-s − 0.929·13-s − 0.168i·14-s + (−0.192 + 0.102i)15-s + 0.250·16-s − 0.753·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.335 - 0.942i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.335 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.875270 + 0.617346i\)
\(L(\frac12)\) \(\approx\) \(0.875270 + 0.617346i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 - 1.97i)T \)
37 \( 1 + (-4.10 - 4.48i)T \)
good3 \( 1 - 0.377iT - 3T^{2} \)
7 \( 1 - 0.631iT - 7T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 + 3.34T + 13T^{2} \)
17 \( 1 + 3.10T + 17T^{2} \)
19 \( 1 - 5.97iT - 19T^{2} \)
23 \( 1 - 7.60T + 23T^{2} \)
29 \( 1 + 9.57iT - 29T^{2} \)
31 \( 1 - 7.26iT - 31T^{2} \)
41 \( 1 + 8.45T + 41T^{2} \)
43 \( 1 - 4.86T + 43T^{2} \)
47 \( 1 - 13.1iT - 47T^{2} \)
53 \( 1 + 7.17iT - 53T^{2} \)
59 \( 1 + 4.36iT - 59T^{2} \)
61 \( 1 + 2.14iT - 61T^{2} \)
67 \( 1 + 11.3iT - 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 4.45iT - 73T^{2} \)
79 \( 1 + 8.78iT - 79T^{2} \)
83 \( 1 + 6.63iT - 83T^{2} \)
89 \( 1 + 13.7iT - 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−11.34865082008331595081806445919, −10.41011204515292485205854789216, −9.841508251892304469620079044189, −9.051663779321409187700004005937, −7.74834798322188401603067994271, −6.93649003210927709506697858837, −6.08764115760299229351841198200, −4.63297093511628317213471830938, −3.12463151612735575852697414868, −1.79878270560593480113151527628, 0.986373115977527113288308057356, 2.39325109486815813608967620118, 4.32524178446641243031114995266, 5.31980754578227892791848174868, 6.85406088087051770959402980502, 7.29115353954443194577573826767, 8.750278368409933188403544995033, 9.229424851736031215477023714688, 10.14451642980638934131305221529, 11.09114501568638377112317572186

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