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Properties

Label 2-370-37.16-c1-0-8
Degree $2$
Conductor $370$
Sign $-0.0216 + 0.999i$
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 + (−0.436 + 0.159i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + 0.465·6-s + (0.593 − 3.36i)7-s + (−0.500 − 0.866i)8-s + (−2.13 + 1.78i)9-s + (0.5 − 0.866i)10-s + (−3.22 − 5.58i)11-s + (−0.436 − 0.159i)12-s + (3.08 + 2.58i)13-s + (−1.70 + 2.95i)14-s + (−0.0807 − 0.457i)15-s + (0.173 + 0.984i)16-s + (2.55 − 2.14i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (−0.252 + 0.0918i)3-s + (0.383 + 0.321i)4-s + (−0.0776 + 0.440i)5-s + 0.189·6-s + (0.224 − 1.27i)7-s + (−0.176 − 0.306i)8-s + (−0.710 + 0.596i)9-s + (0.158 − 0.273i)10-s + (−0.972 − 1.68i)11-s + (−0.126 − 0.0459i)12-s + (0.855 + 0.717i)13-s + (−0.456 + 0.790i)14-s + (−0.0208 − 0.118i)15-s + (0.0434 + 0.246i)16-s + (0.620 − 0.520i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0216 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0216 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.0216 + 0.999i$
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: \((2,\ 370,\ (\ :1/2),\ -0.0216 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.502459 - 0.513456i\)
\(L(\frac12)\) \(\approx\) \(0.502459 - 0.513456i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.52 + 4.95i)T \)
good3 \( 1 + (0.436 - 0.159i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.593 + 3.36i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (3.22 + 5.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.08 - 2.58i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.55 + 2.14i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (-1.82 + 0.662i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (-3.97 + 6.88i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.45 + 7.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.45T + 31T^{2} \)
41 \( 1 + (-2.06 - 1.73i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 5.37T + 43T^{2} \)
47 \( 1 + (4.77 - 8.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.523 + 2.96i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (0.145 + 0.824i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.44 - 2.04i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.828 + 4.69i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.83 + 2.12i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + (2.17 - 12.3i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (5.12 - 4.29i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (0.0213 + 0.121i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (3.31 - 5.74i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.87815671284538619134491355795, −10.72699767071200686267306380273, −9.366696161968068302295114416645, −8.255110136478101886656921705140, −7.66097932309995832100064516800, −6.48813069123643767229763002685, −5.39976560098598996307728070234, −3.85762411464769587660311433143, −2.70700301349335643888111249516, −0.63301431574460952776559645198, 1.68004025274204825913817268634, 3.23885727794462141764908250577, 5.36787892550354934939439470301, 5.53698044289636938064350766188, 7.06736238182779155764422799073, 8.014129681662082458346822204060, 8.881702931830910520830450284531, 9.582823777831799778896829743296, 10.67085117084674213856664509511, 11.62426465879838572806151469499

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