Properties

Label 2-370-185.98-c1-0-6
Degree $2$
Conductor $370$
Sign $0.816 - 0.577i$
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  + (0.173 − 0.984i)2-s + (2.40 + 1.68i)3-s + (−0.939 − 0.342i)4-s + (−1.06 + 1.96i)5-s + (2.07 − 2.07i)6-s + (−0.0272 + 0.311i)7-s + (−0.5 + 0.866i)8-s + (1.91 + 5.25i)9-s + (1.75 + 1.39i)10-s + (−0.336 − 0.194i)11-s + (−1.68 − 2.40i)12-s + (1.10 + 0.402i)13-s + (0.302 + 0.0810i)14-s + (−5.86 + 2.92i)15-s + (0.766 + 0.642i)16-s + (0.856 + 2.35i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (1.38 + 0.970i)3-s + (−0.469 − 0.171i)4-s + (−0.476 + 0.878i)5-s + (0.846 − 0.846i)6-s + (−0.0103 + 0.117i)7-s + (−0.176 + 0.306i)8-s + (0.637 + 1.75i)9-s + (0.553 + 0.440i)10-s + (−0.101 − 0.0585i)11-s + (−0.485 − 0.693i)12-s + (0.306 + 0.111i)13-s + (0.0808 + 0.0216i)14-s + (−1.51 + 0.755i)15-s + (0.191 + 0.160i)16-s + (0.207 + 0.570i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85591 + 0.589507i\)
\(L(\frac12)\) \(\approx\) \(1.85591 + 0.589507i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (1.06 - 1.96i)T \)
37 \( 1 + (2.74 + 5.42i)T \)
good3 \( 1 + (-2.40 - 1.68i)T + (1.02 + 2.81i)T^{2} \)
7 \( 1 + (0.0272 - 0.311i)T + (-6.89 - 1.21i)T^{2} \)
11 \( 1 + (0.336 + 0.194i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.10 - 0.402i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.856 - 2.35i)T + (-13.0 + 10.9i)T^{2} \)
19 \( 1 + (-3.50 + 5.00i)T + (-6.49 - 17.8i)T^{2} \)
23 \( 1 + (-0.00670 - 0.0116i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.46 - 5.48i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (1.63 + 1.63i)T + 31iT^{2} \)
41 \( 1 + (-3.16 + 8.68i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + 9.85T + 43T^{2} \)
47 \( 1 + (-3.99 - 1.07i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.404 + 4.61i)T + (-52.1 + 9.20i)T^{2} \)
59 \( 1 + (0.834 + 9.54i)T + (-58.1 + 10.2i)T^{2} \)
61 \( 1 + (2.39 + 5.12i)T + (-39.2 + 46.7i)T^{2} \)
67 \( 1 + (-13.1 - 1.15i)T + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (0.00507 + 0.0287i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (1.83 - 1.83i)T - 73iT^{2} \)
79 \( 1 + (11.5 + 1.01i)T + (77.7 + 13.7i)T^{2} \)
83 \( 1 + (-14.8 - 6.94i)T + (53.3 + 63.5i)T^{2} \)
89 \( 1 + (13.5 - 1.18i)T + (87.6 - 15.4i)T^{2} \)
97 \( 1 + (-7.27 + 4.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.14096519816281935043305099856, −10.59614442874681266514938157591, −9.684844817667789419257797958657, −8.906852036683765541338285505125, −8.069887771048195140302206749151, −6.97467298325803507956032064440, −5.23283982254886326865707491700, −3.95987031530422844671464274503, −3.31606994237690442586863210233, −2.32302948438745481186444938784, 1.29832124835976090046621605304, 3.07700939809215644155499626642, 4.18085115755662750230802197393, 5.56847367836991721718331965273, 6.87552169147214252240203748773, 7.78987360795244196013745015539, 8.213581687806218802025487395148, 9.078648430672247454719146157466, 9.942784306093188101556671948833, 11.79425283407754435961172930427

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