Properties

Label 2-370-185.103-c1-0-0
Degree $2$
Conductor $370$
Sign $0.782 - 0.622i$
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  + (−0.5 − 0.866i)2-s + (−0.758 − 2.83i)3-s + (−0.499 + 0.866i)4-s + (−1.61 + 1.55i)5-s + (−2.07 + 2.07i)6-s + (0.455 + 1.69i)7-s + 0.999·8-s + (−4.84 + 2.79i)9-s + (2.14 + 0.619i)10-s + 5.29i·11-s + (2.83 + 0.758i)12-s + (−0.197 + 0.342i)13-s + (1.24 − 1.24i)14-s + (5.61 + 3.38i)15-s + (−0.5 − 0.866i)16-s + (3.21 − 1.85i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.438 − 1.63i)3-s + (−0.249 + 0.433i)4-s + (−0.720 + 0.693i)5-s + (−0.846 + 0.846i)6-s + (0.172 + 0.641i)7-s + 0.353·8-s + (−1.61 + 0.932i)9-s + (0.679 + 0.195i)10-s + 1.59i·11-s + (0.817 + 0.219i)12-s + (−0.0549 + 0.0951i)13-s + (0.332 − 0.332i)14-s + (1.44 + 0.873i)15-s + (−0.125 − 0.216i)16-s + (0.778 − 0.449i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.782 - 0.622i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.782 - 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.413794 + 0.144520i\)
\(L(\frac12)\) \(\approx\) \(0.413794 + 0.144520i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (1.61 - 1.55i)T \)
37 \( 1 + (-2.51 - 5.53i)T \)
good3 \( 1 + (0.758 + 2.83i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.455 - 1.69i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 5.29iT - 11T^{2} \)
13 \( 1 + (0.197 - 0.342i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.21 + 1.85i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.24 - 1.40i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 7.28T + 23T^{2} \)
29 \( 1 + (1.33 - 1.33i)T - 29iT^{2} \)
31 \( 1 + (-3.75 - 3.75i)T + 31iT^{2} \)
41 \( 1 + (5.24 + 3.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.65T + 43T^{2} \)
47 \( 1 + (7.10 - 7.10i)T - 47iT^{2} \)
53 \( 1 + (2.33 - 8.69i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.49 - 5.59i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.793 - 0.212i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.42 - 0.917i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.22 - 2.12i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.76 + 7.76i)T - 73iT^{2} \)
79 \( 1 + (12.4 - 3.33i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (0.519 - 1.93i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.441 - 0.118i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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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

−11.94940695273580789068744613327, −10.78740239406687491136191841443, −9.846381720196599758995987637397, −8.409271052860105224079369400775, −7.69742751659189513025083334383, −6.98801626431232734999667799679, −6.00310994455906728578054085834, −4.43157388858496600655210564584, −2.69225250759224730327879523127, −1.72285252141415801453902056257, 0.35028194899048106830667011841, 3.67736805921886314042450089189, 4.29068644236957626114046029823, 5.40321827890047302197063494804, 6.21507407166420386614709285147, 7.944034274210518567549897978775, 8.480486517548494220648529488909, 9.467294671685231420836222058067, 10.37007662897974730322557570390, 11.04006297792805229820413319053

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