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Properties

Label 2-370-185.9-c1-0-12
Degree $2$
Conductor $370$
Sign $0.934 - 0.355i$
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  + (−0.984 + 0.173i)2-s + (2.61 + 0.460i)3-s + (0.939 − 0.342i)4-s + (1.38 + 1.75i)5-s − 2.65·6-s + (2.09 − 2.49i)7-s + (−0.866 + 0.5i)8-s + (3.79 + 1.38i)9-s + (−1.66 − 1.48i)10-s + (0.0622 + 0.107i)11-s + (2.61 − 0.460i)12-s + (−2.13 − 5.85i)13-s + (−1.62 + 2.81i)14-s + (2.81 + 5.22i)15-s + (0.766 − 0.642i)16-s + (−1.61 + 4.42i)17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (1.50 + 0.265i)3-s + (0.469 − 0.171i)4-s + (0.619 + 0.784i)5-s − 1.08·6-s + (0.790 − 0.941i)7-s + (−0.306 + 0.176i)8-s + (1.26 + 0.460i)9-s + (−0.527 − 0.470i)10-s + (0.0187 + 0.0324i)11-s + (0.754 − 0.132i)12-s + (−0.591 − 1.62i)13-s + (−0.434 + 0.753i)14-s + (0.726 + 1.34i)15-s + (0.191 − 0.160i)16-s + (−0.390 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.934 - 0.355i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.934 - 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75895 + 0.323265i\)
\(L(\frac12)\) \(\approx\) \(1.75895 + 0.323265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
5 \( 1 + (-1.38 - 1.75i)T \)
37 \( 1 + (-2.12 - 5.69i)T \)
good3 \( 1 + (-2.61 - 0.460i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (-2.09 + 2.49i)T + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.0622 - 0.107i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.13 + 5.85i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.61 - 4.42i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.307 - 1.74i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (7.34 + 4.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.35T + 31T^{2} \)
41 \( 1 + (3.81 - 1.38i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + 2.22iT - 43T^{2} \)
47 \( 1 + (-8.31 - 4.80i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.66 - 5.56i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-5.60 + 4.70i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.127 + 0.0462i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (6.98 - 8.32i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.01 + 5.74i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + 9.15iT - 73T^{2} \)
79 \( 1 + (-0.898 - 0.754i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.33 - 3.65i)T + (-63.5 - 53.3i)T^{2} \)
89 \( 1 + (11.3 - 9.50i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (1.40 + 0.809i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.81744315302525767448282741697, −10.35952803550849820534173664382, −9.732169199813271068089074807305, −8.465249644412618937050174093366, −7.950642644941947761303482891098, −7.15628505838423070672434262261, −5.81556043884772754359266464452, −4.13449708787836332660187771950, −2.94532079903748788895229529376, −1.84178081612819763346637083878, 1.91640039576920458192777725393, 2.32919952283459082063402924067, 4.16490717381568437128627179879, 5.49602466657445896487993514239, 6.99450601257470302379000467820, 7.918513146723409453580539755628, 8.880875697025969321394912295271, 9.125455717099209767866990462599, 9.897845851207722302704265735314, 11.55669561996159780237266505540

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