Properties

Label 2-370-37.34-c1-0-3
Degree $2$
Conductor $370$
Sign $-0.651 - 0.758i$
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 + (−0.646 + 0.542i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s − 0.843·6-s + (0.282 + 0.102i)7-s + (−0.500 + 0.866i)8-s + (−0.397 + 2.25i)9-s + (−0.5 − 0.866i)10-s + (−3.09 + 5.35i)11-s + (−0.646 − 0.542i)12-s + (−0.104 − 0.593i)13-s + (0.150 + 0.260i)14-s + (0.792 − 0.288i)15-s + (−0.939 + 0.342i)16-s + (−0.274 + 1.55i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (−0.373 + 0.313i)3-s + (0.0868 + 0.492i)4-s + (−0.420 − 0.152i)5-s − 0.344·6-s + (0.106 + 0.0388i)7-s + (−0.176 + 0.306i)8-s + (−0.132 + 0.751i)9-s + (−0.158 − 0.273i)10-s + (−0.931 + 1.61i)11-s + (−0.186 − 0.156i)12-s + (−0.0290 − 0.164i)13-s + (0.0401 + 0.0694i)14-s + (0.204 − 0.0744i)15-s + (−0.234 + 0.0855i)16-s + (−0.0665 + 0.377i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.651 - 0.758i$
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: \((2,\ 370,\ (\ :1/2),\ -0.651 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515395 + 1.12158i\)
\(L(\frac12)\) \(\approx\) \(0.515395 + 1.12158i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.42 + 2.74i)T \)
good3 \( 1 + (0.646 - 0.542i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.282 - 0.102i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (3.09 - 5.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.104 + 0.593i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.274 - 1.55i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (-3.68 + 3.08i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-3.32 - 5.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0493 - 0.0854i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.46T + 31T^{2} \)
41 \( 1 + (1.46 + 8.30i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 - 5.55T + 43T^{2} \)
47 \( 1 + (-5.91 - 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-12.8 + 4.67i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-10.9 + 3.99i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.63 - 9.26i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.75 - 1.36i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (6.27 - 5.26i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + 4.14T + 73T^{2} \)
79 \( 1 + (-6.38 - 2.32i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.46 + 8.28i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-5.34 + 1.94i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (5.32 + 9.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.72034368713217257688563512033, −10.96332212724695811211319925186, −9.997422551434000539381435928344, −8.900618753448571667781598854624, −7.45546338870212956669625181474, −7.39015543755081615388653224409, −5.53135671272530963189853856109, −5.04165650181581730654118454930, −3.96429547386679690307913856111, −2.35841535030215127547625688334, 0.73898235793506164080505871052, 2.82761269349839925751937545859, 3.78208562158474579406652551670, 5.25467663417446560403453324487, 6.03845787552011003595392771348, 7.12373351135358170124798664478, 8.265075669054989208802623200159, 9.282814305207035389162202519362, 10.54101398948912199734706154489, 11.22732607513294963347052209356

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