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Properties

Label 2-370-37.33-c1-0-8
Degree $2$
Conductor $370$
Sign $0.647 - 0.762i$
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 + (0.160 + 0.909i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s − 0.923·6-s + (1.63 − 1.36i)7-s + (0.5 − 0.866i)8-s + (2.01 − 0.734i)9-s + (0.5 + 0.866i)10-s + (1.53 − 2.65i)11-s + (0.160 − 0.909i)12-s + (3.64 + 1.32i)13-s + (1.06 + 1.84i)14-s + (0.707 + 0.593i)15-s + (0.766 + 0.642i)16-s + (−6.15 + 2.24i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (0.0926 + 0.525i)3-s + (−0.469 − 0.171i)4-s + (0.342 − 0.287i)5-s − 0.377·6-s + (0.616 − 0.517i)7-s + (0.176 − 0.306i)8-s + (0.672 − 0.244i)9-s + (0.158 + 0.273i)10-s + (0.461 − 0.799i)11-s + (0.0463 − 0.262i)12-s + (1.01 + 0.367i)13-s + (0.284 + 0.492i)14-s + (0.182 + 0.153i)15-s + (0.191 + 0.160i)16-s + (−1.49 + 0.543i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.647 - 0.762i$
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: \((2,\ 370,\ (\ :1/2),\ 0.647 - 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35925 + 0.629195i\)
\(L(\frac12)\) \(\approx\) \(1.35925 + 0.629195i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.88 - 5.35i)T \)
good3 \( 1 + (-0.160 - 0.909i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-1.63 + 1.36i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-1.53 + 2.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.64 - 1.32i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (6.15 - 2.24i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-0.665 - 3.77i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-0.436 - 0.756i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.64 + 8.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
41 \( 1 + (-7.73 - 2.81i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 - 5.01T + 43T^{2} \)
47 \( 1 + (3.62 + 6.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.36 - 7.86i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (8.59 + 7.21i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (11.9 + 4.35i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.31 - 1.10i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.0544 - 0.308i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + (-3.39 + 2.84i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (12.0 - 4.38i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (5.18 + 4.35i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-2.24 - 3.88i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.22096191074398343765849752592, −10.59286822728426511444789478346, −9.457848055555546105203163439135, −8.788996042922655625871854173732, −7.889583927873017519443865971134, −6.65388983081448482799877484582, −5.85845362024801948850856097001, −4.47444807112361048038112048410, −3.83805553837700737030721330397, −1.43947837897745670177523742393, 1.53196611033486691409749194640, 2.53269122436118956525690188157, 4.16471887640298907947326008150, 5.23849438987547569537089656789, 6.68686166528963314298986609155, 7.46536750414521381688537724224, 8.796468702018735865709330124781, 9.282563762946823406298716627170, 10.70118414079811269548107049053, 11.06232355869127382254357403905

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