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Properties

Label 2-370-185.64-c1-0-5
Degree $2$
Conductor $370$
Sign $-0.269 - 0.963i$
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 + (0.802 − 0.463i)3-s + (−0.499 − 0.866i)4-s + (1.82 + 1.29i)5-s + 0.926i·6-s + (−3.13 + 1.81i)7-s + 0.999·8-s + (−1.07 + 1.85i)9-s + (−2.03 + 0.927i)10-s − 2.25·11-s + (−0.802 − 0.463i)12-s + (2.51 + 4.35i)13-s − 3.62i·14-s + (2.06 + 0.198i)15-s + (−0.5 + 0.866i)16-s + (−0.941 + 1.63i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.463 − 0.267i)3-s + (−0.249 − 0.433i)4-s + (0.814 + 0.580i)5-s + 0.378i·6-s + (−1.18 + 0.684i)7-s + 0.353·8-s + (−0.356 + 0.618i)9-s + (−0.643 + 0.293i)10-s − 0.680·11-s + (−0.231 − 0.133i)12-s + (0.697 + 1.20i)13-s − 0.967i·14-s + (0.532 + 0.0513i)15-s + (−0.125 + 0.216i)16-s + (−0.228 + 0.395i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.269 - 0.963i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.269 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.708738 + 0.934016i\)
\(L(\frac12)\) \(\approx\) \(0.708738 + 0.934016i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 - 1.29i)T \)
37 \( 1 + (5.63 + 2.28i)T \)
good3 \( 1 + (-0.802 + 0.463i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (3.13 - 1.81i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.25T + 11T^{2} \)
13 \( 1 + (-2.51 - 4.35i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.941 - 1.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.77 + 2.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.62T + 23T^{2} \)
29 \( 1 + 0.243iT - 29T^{2} \)
31 \( 1 + 6.15iT - 31T^{2} \)
41 \( 1 + (-4.85 - 8.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.54T + 43T^{2} \)
47 \( 1 + 4.91iT - 47T^{2} \)
53 \( 1 + (10.6 + 6.13i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.09 + 0.632i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.70 + 3.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.14 - 2.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.93 - 5.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.0iT - 73T^{2} \)
79 \( 1 + (-9.17 + 5.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.77 - 4.49i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-11.0 - 6.36i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−11.46312175413653016482309353126, −10.59910703334362560636700588962, −9.424580769714848574604809163205, −9.119195957732137111138717086286, −7.902506057868903895664125606873, −6.84152151769733278480487370208, −6.13641646493057642041882691078, −5.15152996193828289749833758481, −3.20398201121182160400072766784, −2.12015564132666853463154167354, 0.861956367485133803003959960056, 2.88010615724883963252034887804, 3.57375781719374700102746227210, 5.18766738844080625859849604100, 6.26477965505441212617641921376, 7.57582018722536575493252553806, 8.717459639082744788944389448608, 9.378273201194547711938679348488, 10.15013281720269900402344116831, 10.75282580914825894714083370846

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