Properties

Label 2-370-37.34-c1-0-11
Degree $2$
Conductor $370$
Sign $0.999 + 0.0301i$
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  + (0.766 + 0.642i)2-s + (1.78 − 1.49i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + 2.32·6-s + (2.45 + 0.892i)7-s + (−0.500 + 0.866i)8-s + (0.420 − 2.38i)9-s + (−0.5 − 0.866i)10-s + (0.947 − 1.64i)11-s + (1.78 + 1.49i)12-s + (−0.156 − 0.888i)13-s + (1.30 + 2.26i)14-s + (−2.18 + 0.796i)15-s + (−0.939 + 0.342i)16-s + (0.126 − 0.719i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (1.02 − 0.864i)3-s + (0.0868 + 0.492i)4-s + (−0.420 − 0.152i)5-s + 0.950·6-s + (0.926 + 0.337i)7-s + (−0.176 + 0.306i)8-s + (0.140 − 0.794i)9-s + (−0.158 − 0.273i)10-s + (0.285 − 0.494i)11-s + (0.514 + 0.432i)12-s + (−0.0434 − 0.246i)13-s + (0.348 + 0.604i)14-s + (−0.564 + 0.205i)15-s + (−0.234 + 0.0855i)16-s + (0.0307 − 0.174i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.999 + 0.0301i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.999 + 0.0301i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41848 - 0.0364386i\)
\(L(\frac12)\) \(\approx\) \(2.41848 - 0.0364386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (1.62 - 5.86i)T \)
good3 \( 1 + (-1.78 + 1.49i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-2.45 - 0.892i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.947 + 1.64i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.156 + 0.888i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.126 + 0.719i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (3.60 - 3.02i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (0.507 + 0.878i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.79 + 3.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.42T + 31T^{2} \)
41 \( 1 + (1.19 + 6.76i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 4.52T + 43T^{2} \)
47 \( 1 + (-2.52 - 4.37i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.31 - 2.29i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (2.99 - 1.09i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.39 - 13.5i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.82 + 2.11i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-3.08 + 2.58i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + 3.69T + 73T^{2} \)
79 \( 1 + (-5.57 - 2.02i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.504 + 2.85i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-2.41 + 0.879i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (4.45 + 7.71i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.73926366614934597654700593840, −10.65283435861240797567701985929, −9.031640503653718318052190402795, −8.293538165758619485976567640223, −7.81270052644826495299734946965, −6.79248889041744711974678254531, −5.60547422328103183400087030428, −4.35344950482049132472604435100, −3.11449101275951972469218514105, −1.80135731958188192620457014763, 1.98415295208640506950617574637, 3.35910355181060520090945020578, 4.24964466117240345017933949508, 4.97473023797430231957050061391, 6.67261984403403579057988387162, 7.83701882252447727998082643836, 8.775288868198557915969473873227, 9.602159911599338348966418442376, 10.61143739203742810583850219102, 11.24192001063988654271658581085

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