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Properties

Label 2-370-37.9-c1-0-12
Degree $2$
Conductor $370$
Sign $-0.999 - 0.0324i$
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 + (0.550 − 3.12i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s − 3.17·6-s + (0.712 + 0.597i)7-s + (0.5 + 0.866i)8-s + (−6.63 − 2.41i)9-s + (0.5 − 0.866i)10-s + (−2.93 − 5.08i)11-s + (0.550 + 3.12i)12-s + (−3.32 + 1.20i)13-s + (0.464 − 0.804i)14-s + (2.42 − 2.03i)15-s + (0.766 − 0.642i)16-s + (1.16 + 0.424i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (0.317 − 1.80i)3-s + (−0.469 + 0.171i)4-s + (0.342 + 0.287i)5-s − 1.29·6-s + (0.269 + 0.225i)7-s + (0.176 + 0.306i)8-s + (−2.21 − 0.804i)9-s + (0.158 − 0.273i)10-s + (−0.885 − 1.53i)11-s + (0.158 + 0.901i)12-s + (−0.921 + 0.335i)13-s + (0.124 − 0.215i)14-s + (0.627 − 0.526i)15-s + (0.191 − 0.160i)16-s + (0.282 + 0.102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.999 - 0.0324i$
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: \((2,\ 370,\ (\ :1/2),\ -0.999 - 0.0324i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0193486 + 1.19348i\)
\(L(\frac12)\) \(\approx\) \(0.0193486 + 1.19348i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.90 - 1.46i)T \)
good3 \( 1 + (-0.550 + 3.12i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (-0.712 - 0.597i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (2.93 + 5.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.32 - 1.20i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.16 - 0.424i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (-0.326 + 1.85i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (-3.02 + 5.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.01 - 5.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.74T + 31T^{2} \)
41 \( 1 + (-3.88 + 1.41i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + 2.60T + 43T^{2} \)
47 \( 1 + (1.21 - 2.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.31 + 3.61i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-9.34 + 7.83i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.0319 + 0.0116i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-10.2 - 8.56i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.799 - 4.53i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
79 \( 1 + (6.22 + 5.22i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (5.44 + 1.98i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-7.69 + 6.46i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (5.43 - 9.41i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.24444397250681098021482104856, −10.15619158789341663703958462928, −8.743066477091460761255325740329, −8.279822112768964527713308735507, −7.23197438407547441741712737381, −6.27839061955544455687007392600, −5.17054773697142816210880622102, −2.99407867277225693183185276953, −2.37999690021676019022936189683, −0.820330536698781329594248487736, 2.68931837337007468032301719938, 4.29434887602811862666532126785, 4.88726727157015243663901015930, 5.69468713756202973176737490935, 7.41574755110730115012467491756, 8.186519447583189377253558880874, 9.385403722472648858091804207709, 9.907604295662756063204840163080, 10.38388702737137176518476317623, 11.67660610955331926287471405812

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