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Properties

Label 2-370-37.9-c1-0-8
Degree $2$
Conductor $370$
Sign $0.972 + 0.233i$
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  + (0.173 + 0.984i)2-s + (0.357 − 2.02i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + 2.06·6-s + (3.79 + 3.18i)7-s + (−0.5 − 0.866i)8-s + (−1.17 − 0.425i)9-s + (0.5 − 0.866i)10-s + (−2.17 − 3.76i)11-s + (0.357 + 2.02i)12-s + (5.57 − 2.03i)13-s + (−2.48 + 4.29i)14-s + (−1.57 + 1.32i)15-s + (0.766 − 0.642i)16-s + (−2.71 − 0.986i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (0.206 − 1.17i)3-s + (−0.469 + 0.171i)4-s + (−0.342 − 0.287i)5-s + 0.841·6-s + (1.43 + 1.20i)7-s + (−0.176 − 0.306i)8-s + (−0.390 − 0.141i)9-s + (0.158 − 0.273i)10-s + (−0.655 − 1.13i)11-s + (0.103 + 0.585i)12-s + (1.54 − 0.563i)13-s + (−0.662 + 1.14i)14-s + (−0.407 + 0.341i)15-s + (0.191 − 0.160i)16-s + (−0.657 − 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.972 + 0.233i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.972 + 0.233i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59249 - 0.188786i\)
\(L(\frac12)\) \(\approx\) \(1.59249 - 0.188786i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (5.48 + 2.62i)T \)
good3 \( 1 + (-0.357 + 2.02i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (-3.79 - 3.18i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (2.17 + 3.76i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.57 + 2.03i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.71 + 0.986i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (-0.0838 + 0.475i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (-0.285 + 0.493i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.33 - 4.05i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
41 \( 1 + (4.46 - 1.62i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + 7.15T + 43T^{2} \)
47 \( 1 + (4.54 - 7.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.818 + 0.686i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (0.371 - 0.311i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (9.26 - 3.37i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.319 - 0.268i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.70 - 9.69i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + (-2.99 - 2.51i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-8.20 - 2.98i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-8.28 + 6.95i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (5.85 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.57516390373248329314216873062, −10.71334423241538933648543677034, −8.747101413989454994045387313122, −8.388304820493634060237567386033, −7.890312610499393610890449836698, −6.56184994831481163283258464268, −5.68699589607532630530087893850, −4.72735971926928906463913969313, −2.92108781010185063800153579364, −1.30971096801341615612378651079, 1.67868390757737347605074098065, 3.51778375281730073469174600159, 4.41863933422188177001125722997, 4.82912209588873553587152672570, 6.72436855249539810049174549518, 7.985697144773787228731432779615, 8.728638084533503291261554861455, 10.10597655035962355774815488613, 10.43179027435769399511645379493, 11.21582435318506302436219381229

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