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Properties

Label 2-370-185.64-c1-0-8
Degree $2$
Conductor $370$
Sign $0.443 + 0.896i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−2.18 + 1.26i)3-s + (−0.499 − 0.866i)4-s + (−1.5 + 1.65i)5-s − 2.52i·6-s + (−3 + 1.73i)7-s + 0.999·8-s + (1.68 − 2.92i)9-s + (−0.686 − 2.12i)10-s + (2.18 + 1.26i)12-s + (0.186 + 0.322i)13-s − 3.46i·14-s + (1.18 − 5.51i)15-s + (−0.5 + 0.866i)16-s + (0.686 − 1.18i)17-s + (1.68 + 2.92i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−1.26 + 0.728i)3-s + (−0.249 − 0.433i)4-s + (−0.670 + 0.741i)5-s − 1.03i·6-s + (−1.13 + 0.654i)7-s + 0.353·8-s + (0.562 − 0.973i)9-s + (−0.216 − 0.672i)10-s + (0.631 + 0.364i)12-s + (0.0516 + 0.0894i)13-s − 0.925i·14-s + (0.306 − 1.42i)15-s + (−0.125 + 0.216i)16-s + (0.166 − 0.288i)17-s + (0.397 + 0.688i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.443 + 0.896i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.443 + 0.896i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 1.65i)T \)
37 \( 1 + (-0.5 + 6.06i)T \)
good3 \( 1 + (2.18 - 1.26i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (3 - 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-0.186 - 0.322i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.686 + 1.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.74T + 23T^{2} \)
29 \( 1 + 7.72iT - 29T^{2} \)
31 \( 1 - 11.0iT - 31T^{2} \)
41 \( 1 + (-2.87 - 4.97i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 9.11T + 43T^{2} \)
47 \( 1 + 5.04iT - 47T^{2} \)
53 \( 1 + (-0.813 - 0.469i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.3 + 5.98i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.05 + 1.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.11 - 4.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.37 + 2.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + (7.11 - 4.10i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.7 + 6.78i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.31 + 4.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−11.15096555550551929826484276646, −10.25689682820376936415781458243, −9.628035611388894118883215687959, −8.495237136544163940562244010178, −7.14638913893938203619892257692, −6.39664827871028382805597708233, −5.60981092847855310671001524995, −4.46377013943940548994635233799, −3.12616451694725125542948197612, 0, 1.19940740588260723906817380778, 3.36293840380481594562593739434, 4.59102875602102777040626949036, 5.82907691661979055338889169674, 6.89633244033457933420439212749, 7.68156597875810815462165669009, 8.894512461875042411245339024924, 9.917858083197367801430204768553, 10.86685320512538130974659479043

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