Properties

Label 2-370-37.16-c1-0-0
Degree $2$
Conductor $370$
Sign $-0.666 - 0.745i$
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  + (−0.939 − 0.342i)2-s + (−0.142 + 0.0519i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + 0.151·6-s + (−0.282 + 1.60i)7-s + (−0.500 − 0.866i)8-s + (−2.28 + 1.91i)9-s + (0.5 − 0.866i)10-s + (0.316 + 0.548i)11-s + (−0.142 − 0.0519i)12-s + (−4.82 − 4.04i)13-s + (0.812 − 1.40i)14-s + (−0.0263 − 0.149i)15-s + (0.173 + 0.984i)16-s + (−3.18 + 2.67i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.0823 + 0.0299i)3-s + (0.383 + 0.321i)4-s + (−0.0776 + 0.440i)5-s + 0.0619·6-s + (−0.106 + 0.604i)7-s + (−0.176 − 0.306i)8-s + (−0.760 + 0.637i)9-s + (0.158 − 0.273i)10-s + (0.0955 + 0.165i)11-s + (−0.0411 − 0.0149i)12-s + (−1.33 − 1.12i)13-s + (0.217 − 0.376i)14-s + (−0.00680 − 0.0386i)15-s + (0.0434 + 0.246i)16-s + (−0.772 + 0.647i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.666 - 0.745i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.666 - 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175573 + 0.392376i\)
\(L(\frac12)\) \(\approx\) \(0.175573 + 0.392376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (-5.96 - 1.21i)T \)
good3 \( 1 + (0.142 - 0.0519i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (0.282 - 1.60i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-0.316 - 0.548i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.82 + 4.04i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (3.18 - 2.67i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (-0.671 + 0.244i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (1.23 - 2.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.59 - 6.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
41 \( 1 + (-4.87 - 4.09i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + 9.46T + 43T^{2} \)
47 \( 1 + (3.91 - 6.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.50 + 8.54i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-1.72 - 9.79i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-8.15 - 6.84i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-1.61 + 9.18i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (0.643 - 0.234i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 - 3.87T + 73T^{2} \)
79 \( 1 + (-2.94 + 16.6i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.82 + 1.52i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (1.01 + 5.73i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (4.51 - 7.82i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.52391754044354602858915969039, −10.77878719854435763134784788165, −9.974025504530376976393261762940, −8.996979505095229971215921839714, −8.063223791128138770618689656817, −7.25441935865074715383413008618, −6.00877427529157958914980861207, −4.97371249338161628086415442796, −3.19126973246021427268504099806, −2.20832938150706804737487617789, 0.33552395107614820252390234047, 2.31753946135036266079862279951, 4.03741781902994917837392453598, 5.25075072993877896440741366929, 6.52718188776360890198109849422, 7.24297872435283573587163055929, 8.371766139274018124594989188440, 9.287847388800424396065881537097, 9.835816982196837887625534815434, 11.15907990608327170070535514278

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