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Properties

Label 2-370-37.33-c1-0-10
Degree $2$
Conductor $370$
Sign $-0.116 + 0.993i$
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 + (−0.0317 − 0.180i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s − 0.182·6-s + (1.13 − 0.952i)7-s + (−0.5 + 0.866i)8-s + (2.78 − 1.01i)9-s + (−0.5 − 0.866i)10-s + (−0.0395 + 0.0685i)11-s + (−0.0317 + 0.180i)12-s + (−1.35 − 0.492i)13-s + (−0.740 − 1.28i)14-s + (−0.140 − 0.117i)15-s + (0.766 + 0.642i)16-s + (−0.0477 + 0.0173i)17-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.0183 − 0.103i)3-s + (−0.469 − 0.171i)4-s + (0.342 − 0.287i)5-s − 0.0746·6-s + (0.428 − 0.359i)7-s + (−0.176 + 0.306i)8-s + (0.929 − 0.338i)9-s + (−0.158 − 0.273i)10-s + (−0.0119 + 0.0206i)11-s + (−0.00916 + 0.0519i)12-s + (−0.375 − 0.136i)13-s + (−0.197 − 0.342i)14-s + (−0.0361 − 0.0303i)15-s + (0.191 + 0.160i)16-s + (−0.0115 + 0.00421i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.116 + 0.993i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.116 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00347 - 1.12751i\)
\(L(\frac12)\) \(\approx\) \(1.00347 - 1.12751i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.570 - 6.05i)T \)
good3 \( 1 + (0.0317 + 0.180i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-1.13 + 0.952i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.0395 - 0.0685i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.35 + 0.492i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.0477 - 0.0173i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (0.860 + 4.88i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (1.14 + 1.98i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.47 + 2.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.605T + 31T^{2} \)
41 \( 1 + (-6.43 - 2.34i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 8.69T + 43T^{2} \)
47 \( 1 + (-4.53 - 7.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.793 - 0.666i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.696 - 0.584i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-9.99 - 3.63i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.93 - 1.62i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.29 - 7.31i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + 9.62T + 73T^{2} \)
79 \( 1 + (5.04 - 4.23i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.685 + 0.249i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-10.8 - 9.12i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.01 - 1.76i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.17548876339101993713858922629, −10.22012373113561550703668906099, −9.564004969291127268668275112658, −8.523808777641411897651123881498, −7.40851539654391493911588008009, −6.31451563799491957994857960335, −4.91621132159683937842792064141, −4.17949693229827838267375571839, −2.58768679108223922831209188929, −1.13153058792138832002442735000, 1.95561494730454181526005157334, 3.74992986288667009649288965967, 4.91495109703447278469413663178, 5.83342810197030291726462860439, 6.95476654406028268859606967103, 7.76284384074064649474924842016, 8.767230320263983007687757460229, 9.831854575779251434486032963304, 10.50809101865226951607667167036, 11.74531588741692889333949423121

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