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Properties

Label 2-370-185.164-c1-0-9
Degree $2$
Conductor $370$
Sign $0.894 - 0.446i$
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  + (−0.342 + 0.939i)2-s + (0.0373 + 0.102i)3-s + (−0.766 − 0.642i)4-s + (2.23 + 0.0665i)5-s − 0.109·6-s + (0.581 + 0.102i)7-s + (0.866 − 0.500i)8-s + (2.28 − 1.92i)9-s + (−0.826 + 2.07i)10-s + (−1.96 − 3.39i)11-s + (0.0373 − 0.102i)12-s + (1.99 − 2.37i)13-s + (−0.295 + 0.511i)14-s + (0.0766 + 0.231i)15-s + (0.173 + 0.984i)16-s + (3.62 + 4.31i)17-s + ⋯
L(s)  = 1  + (−0.241 + 0.664i)2-s + (0.0215 + 0.0592i)3-s + (−0.383 − 0.321i)4-s + (0.999 + 0.0297i)5-s − 0.0445·6-s + (0.219 + 0.0387i)7-s + (0.306 − 0.176i)8-s + (0.763 − 0.640i)9-s + (−0.261 + 0.656i)10-s + (−0.590 − 1.02i)11-s + (0.0107 − 0.0296i)12-s + (0.553 − 0.659i)13-s + (−0.0789 + 0.136i)14-s + (0.0197 + 0.0598i)15-s + (0.0434 + 0.246i)16-s + (0.878 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.894 - 0.446i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.894 - 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40531 + 0.331555i\)
\(L(\frac12)\) \(\approx\) \(1.40531 + 0.331555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 - 0.939i)T \)
5 \( 1 + (-2.23 - 0.0665i)T \)
37 \( 1 + (3.49 - 4.97i)T \)
good3 \( 1 + (-0.0373 - 0.102i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (-0.581 - 0.102i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (1.96 + 3.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.99 + 2.37i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-3.62 - 4.31i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (5.68 - 2.06i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (-1.90 - 1.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.19 + 3.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.95T + 31T^{2} \)
41 \( 1 + (-7.81 - 6.56i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 8.21iT - 43T^{2} \)
47 \( 1 + (6.59 + 3.80i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (13.1 - 2.32i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (1.92 + 10.9i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.40 - 1.18i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-6.59 - 1.16i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (11.0 - 4.00i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 7.36iT - 73T^{2} \)
79 \( 1 + (0.846 - 4.80i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (4.36 + 5.20i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (1.50 + 8.55i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.416 + 0.240i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.20235352404529214008417152142, −10.26728381921880958456363980983, −9.730428798851341370954411205269, −8.483967159183603529222547369011, −7.946683718358813408823847178225, −6.29753615733219884885101237810, −6.07509358059601979398122843214, −4.75243986504502589918990898362, −3.27570225793539475797009587464, −1.35680850609497394209276104492, 1.60897807368649970490765571371, 2.60018759313719740436615985777, 4.40112456615151621594272239874, 5.19880291082154088969640536518, 6.69199174918373360795467151290, 7.61011959399034174758414928031, 8.822581246136311258723773991351, 9.657791898101999294492667847905, 10.40818465113975754895326081182, 11.07356181145991105645958013929

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