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Properties

Label 2-370-185.174-c1-0-6
Degree $2$
Conductor $370$
Sign $0.999 + 0.0176i$
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  + (−0.866 + 0.5i)2-s + (1.73 + i)3-s + (0.499 − 0.866i)4-s + (−0.469 − 2.18i)5-s − 1.99·6-s + (−2.00 − 1.15i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (1.5 + 1.65i)10-s + 4.31·11-s + (1.73 − 0.999i)12-s + (3.46 + 2i)13-s + 2.31·14-s + (1.37 − 4.25i)15-s + (−0.5 − 0.866i)16-s + (4.60 − 2.65i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.999 + 0.577i)3-s + (0.249 − 0.433i)4-s + (−0.210 − 0.977i)5-s − 0.816·6-s + (−0.758 − 0.437i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.474 + 0.524i)10-s + 1.30·11-s + (0.499 − 0.288i)12-s + (0.960 + 0.554i)13-s + 0.619·14-s + (0.354 − 1.09i)15-s + (−0.125 − 0.216i)16-s + (1.11 − 0.644i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.999 + 0.0176i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.999 + 0.0176i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36017 - 0.0120234i\)
\(L(\frac12)\) \(\approx\) \(1.36017 - 0.0120234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.469 + 2.18i)T \)
37 \( 1 + (2.59 + 5.5i)T \)
good3 \( 1 + (-1.73 - i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.00 + 1.15i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.31T + 11T^{2} \)
13 \( 1 + (-3.46 - 2i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.60 + 2.65i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.15 + 3.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 3.31T + 29T^{2} \)
31 \( 1 - 2.31T + 31T^{2} \)
41 \( 1 + (3.81 - 6.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 8.94iT - 43T^{2} \)
47 \( 1 - 4.31iT - 47T^{2} \)
53 \( 1 + (7.47 - 4.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.31 - 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.97 - 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.45 + 0.841i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.84 + 6.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.63iT - 73T^{2} \)
79 \( 1 + (5.47 - 9.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.74 - 3.31i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.81 - 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.94iT - 97T^{2} \)
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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

−11.38544330640393534361695072213, −9.952291038867924861588913046460, −9.212150656755152649810974912556, −9.019984415444515232013012993266, −7.86193844820896412329993635786, −6.86141110312329257409243509632, −5.63156509199748977107895235462, −4.14559257955724824228646713403, −3.33565392819426832011773052291, −1.21559847215003377504620202141, 1.66496182577626167655062952994, 3.11902955785844298031534569268, 3.58839257132454997498050742682, 6.05782097701599438145989733707, 6.79529275234588770383045645991, 7.940950635860139183284263065359, 8.487414326109150344866705921436, 9.558297007658120442965362392440, 10.32473377963936089324289239450, 11.36036373206994476365029544025

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