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Properties

Label 2-370-37.25-c1-0-5
Degree $2$
Conductor $370$
Sign $0.984 + 0.173i$
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  + (0.642 + 0.766i)2-s + (−1.89 − 1.58i)3-s + (−0.173 + 0.984i)4-s + (0.342 + 0.939i)5-s − 2.46i·6-s + (2.50 − 0.911i)7-s + (−0.866 + 0.500i)8-s + (0.537 + 3.04i)9-s + (−0.5 + 0.866i)10-s + (−1.28 − 2.23i)11-s + (1.89 − 1.58i)12-s + (2.22 + 0.392i)13-s + (2.30 + 1.33i)14-s + (0.844 − 2.31i)15-s + (−0.939 − 0.342i)16-s + (5.64 − 0.995i)17-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (−1.09 − 0.916i)3-s + (−0.0868 + 0.492i)4-s + (0.152 + 0.420i)5-s − 1.00i·6-s + (0.946 − 0.344i)7-s + (−0.306 + 0.176i)8-s + (0.179 + 1.01i)9-s + (−0.158 + 0.273i)10-s + (−0.388 − 0.672i)11-s + (0.545 − 0.458i)12-s + (0.617 + 0.108i)13-s + (0.616 + 0.356i)14-s + (0.217 − 0.598i)15-s + (−0.234 − 0.0855i)16-s + (1.36 − 0.241i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35800 - 0.118739i\)
\(L(\frac12)\) \(\approx\) \(1.35800 - 0.118739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
37 \( 1 + (-0.874 - 6.01i)T \)
good3 \( 1 + (1.89 + 1.58i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-2.50 + 0.911i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (1.28 + 2.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.22 - 0.392i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-5.64 + 0.995i)T + (15.9 - 5.81i)T^{2} \)
19 \( 1 + (-3.50 + 4.17i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (-4.45 - 2.57i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.31 + 0.757i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.01iT - 31T^{2} \)
41 \( 1 + (0.500 - 2.83i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 4.92iT - 43T^{2} \)
47 \( 1 + (5.13 - 8.88i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.2 + 3.72i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.985 - 2.70i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (8.49 + 1.49i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (0.287 - 0.104i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (3.30 + 2.77i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 2.58T + 73T^{2} \)
79 \( 1 + (-3.01 - 8.28i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.982 - 5.57i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (4.18 - 11.4i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-12.7 - 7.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.27686704409387808662934691087, −11.08966033279797379401586592453, −9.549673146426650167878569391221, −8.034577410694112233259722377844, −7.48645307948768043563038789665, −6.48188289016457929769952186142, −5.66129437742363820687891123139, −4.85689670168182003072064548824, −3.15977368784875660623299049689, −1.15026135678703671053536338190, 1.45916777376060705770397715150, 3.43885789833133695254362011277, 4.79421635437769374818552421796, 5.19178079403347332977954137278, 6.09009172323459240331366519103, 7.72730372004526148120791012744, 8.909084815034889076196991544730, 10.05020446112988280212389503195, 10.54071896299092404500637926038, 11.43952723195964215830781836666

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