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Properties

Label 2-370-185.82-c1-0-12
Degree $2$
Conductor $370$
Sign $0.838 + 0.545i$
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  + (0.5 + 0.866i)2-s + (−1.76 + 0.474i)3-s + (−0.499 + 0.866i)4-s + (−0.686 − 2.12i)5-s + (−1.29 − 1.29i)6-s + (1.57 − 0.422i)7-s − 0.999·8-s + (0.308 − 0.178i)9-s + (1.49 − 1.65i)10-s − 5.59i·11-s + (0.474 − 1.76i)12-s + (0.738 − 1.27i)13-s + (1.15 + 1.15i)14-s + (2.22 + 3.44i)15-s + (−0.5 − 0.866i)16-s + (2.61 − 1.50i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−1.02 + 0.273i)3-s + (−0.249 + 0.433i)4-s + (−0.307 − 0.951i)5-s + (−0.528 − 0.528i)6-s + (0.595 − 0.159i)7-s − 0.353·8-s + (0.102 − 0.0593i)9-s + (0.474 − 0.524i)10-s − 1.68i·11-s + (0.136 − 0.510i)12-s + (0.204 − 0.354i)13-s + (0.308 + 0.308i)14-s + (0.574 + 0.888i)15-s + (−0.125 − 0.216i)16-s + (0.633 − 0.366i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.838 + 0.545i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.838 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928809 - 0.275587i\)
\(L(\frac12)\) \(\approx\) \(0.928809 - 0.275587i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.686 + 2.12i)T \)
37 \( 1 + (6.06 + 0.419i)T \)
good3 \( 1 + (1.76 - 0.474i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.57 + 0.422i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + 5.59iT - 11T^{2} \)
13 \( 1 + (-0.738 + 1.27i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.61 + 1.50i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.429 - 1.60i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 7.68T + 23T^{2} \)
29 \( 1 + (7.01 + 7.01i)T + 29iT^{2} \)
31 \( 1 + (-2.74 + 2.74i)T - 31iT^{2} \)
41 \( 1 + (-0.907 - 0.523i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.956T + 43T^{2} \)
47 \( 1 + (0.178 + 0.178i)T + 47iT^{2} \)
53 \( 1 + (-2.63 - 0.705i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (7.07 + 1.89i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.603 + 2.25i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.973 - 3.63i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.226 + 0.392i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.31 + 5.31i)T + 73iT^{2} \)
79 \( 1 + (3.90 + 14.5i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-15.4 - 4.13i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.20 - 8.21i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 - 0.0626iT - 97T^{2} \)
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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

−11.41773833673576306935431710347, −10.72990402410868133191777989251, −9.284093351205456462025962531408, −8.351316299738589371014095868693, −7.64672509161088908450461974627, −6.09363517421215533385821775611, −5.47724810665520920031515539130, −4.72206954587352045737599149976, −3.44179549593405464362399521214, −0.70903972143154312990166212069, 1.67771758226798053082687105140, 3.21456885121322289684034971761, 4.63631658073689559759223842389, 5.48335467600439210577762168188, 6.75831274840853330754672663463, 7.32941254155007029485995480348, 8.882761324574810467856971575763, 10.04936879588344704804087721410, 10.90325904515396328607257212117, 11.40963108394193282691989942099

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