Properties

Label 2-370-37.27-c1-0-3
Degree $2$
Conductor $370$
Sign $0.989 - 0.146i$
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 + (−0.866 − 1.5i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s − 1.73i·6-s + (1.36 + 2.36i)7-s + 0.999i·8-s − 0.999·10-s + 4.73·11-s + (0.866 − 1.49i)12-s + (2.76 − 1.59i)13-s + 2.73i·14-s + (1.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.633 + 0.366i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 − 0.866i)3-s + (0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s − 0.707i·6-s + (0.516 + 0.894i)7-s + 0.353i·8-s − 0.316·10-s + 1.42·11-s + (0.250 − 0.433i)12-s + (0.767 − 0.443i)13-s + 0.730i·14-s + (0.387 + 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.153 + 0.0887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.989 - 0.146i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.989 - 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73704 + 0.127992i\)
\(L(\frac12)\) \(\approx\) \(1.73704 + 0.127992i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.5 - 6.06i)T \)
good3 \( 1 + (0.866 + 1.5i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.36 - 2.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 + (-2.76 + 1.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.633 - 0.366i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.464 + 0.267i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.26iT - 23T^{2} \)
29 \( 1 - 4.92iT - 29T^{2} \)
31 \( 1 + 7.92iT - 31T^{2} \)
41 \( 1 + (-2.23 - 3.86i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.53iT - 43T^{2} \)
47 \( 1 + 6.73T + 47T^{2} \)
53 \( 1 + (6.69 - 11.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.90 + 2.83i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.464 + 0.267i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.19 + 10.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.66T + 73T^{2} \)
79 \( 1 + (-10.3 + 6i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.26 - 7.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.73 + 4.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.3iT - 97T^{2} \)
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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

−11.69486649801268178066438265086, −11.02215442005381349408315732362, −9.392858988123164854260345827050, −8.388129070485076458517396808410, −7.46807023511644957098392711735, −6.40015526654556680904284248397, −5.93794620217405966245438215087, −4.54904320935798573400354869141, −3.28551888779052966624214230986, −1.52340179355847969659158163645, 1.41888258846302516276256559244, 3.73085993349248452966674878099, 4.20813692592435040024906994706, 5.18319793104340302800978732786, 6.40357825123527458600253275971, 7.47451310520575622992258442464, 8.807227524150100610067414657964, 9.804490385378379052637916065755, 10.71758884758770725778505148950, 11.35580350928053526094478606119

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