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Properties

Label 2-370-185.104-c1-0-7
Degree $2$
Conductor $370$
Sign $0.580 + 0.813i$
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  + (−0.939 + 0.342i)2-s + (−0.0975 + 0.267i)3-s + (0.766 − 0.642i)4-s + (−2.23 − 0.121i)5-s − 0.285i·6-s + (−3.86 + 0.681i)7-s + (−0.500 + 0.866i)8-s + (2.23 + 1.87i)9-s + (2.13 − 0.649i)10-s + (2.37 − 4.10i)11-s + (0.0975 + 0.267i)12-s + (3.78 − 3.17i)13-s + (3.39 − 1.96i)14-s + (0.250 − 0.586i)15-s + (0.173 − 0.984i)16-s + (0.839 + 0.704i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−0.0562 + 0.154i)3-s + (0.383 − 0.321i)4-s + (−0.998 − 0.0541i)5-s − 0.116i·6-s + (−1.46 + 0.257i)7-s + (−0.176 + 0.306i)8-s + (0.745 + 0.625i)9-s + (0.676 − 0.205i)10-s + (0.715 − 1.23i)11-s + (0.0281 + 0.0773i)12-s + (1.05 − 0.881i)13-s + (0.908 − 0.524i)14-s + (0.0645 − 0.151i)15-s + (0.0434 − 0.246i)16-s + (0.203 + 0.170i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.580 + 0.813i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.580 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.573024 - 0.295044i\)
\(L(\frac12)\) \(\approx\) \(0.573024 - 0.295044i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (2.23 + 0.121i)T \)
37 \( 1 + (1.98 + 5.74i)T \)
good3 \( 1 + (0.0975 - 0.267i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (3.86 - 0.681i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-2.37 + 4.10i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.78 + 3.17i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.839 - 0.704i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-2.10 + 5.79i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (3.19 + 5.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.58 + 2.64i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.56iT - 31T^{2} \)
41 \( 1 + (6.32 - 5.31i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 9.58T + 43T^{2} \)
47 \( 1 + (-7.68 + 4.43i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.04 - 0.360i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (7.40 + 1.30i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-5.48 - 6.53i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.23 - 0.571i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (13.2 + 4.81i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 2.00iT - 73T^{2} \)
79 \( 1 + (-12.7 + 2.25i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.774 - 0.923i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (2.09 + 0.370i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-5.07 - 8.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.99697441120154127626229571284, −10.43533791530982420889011092465, −9.234154566980578125552944093666, −8.585012311934367399466400536007, −7.56331706235386900738058489466, −6.59049138441800804499479428226, −5.69195905471868281686337108467, −4.01922856354332654356304627904, −3.02540383938928327248472834205, −0.60368203607659798131279375495, 1.43496334104825245265035724596, 3.59451254691994132347341030362, 3.97465962613842656995020779400, 6.17042662098293291889416484584, 7.02060347996509357425849800934, 7.60303104622214761877691140024, 9.061330733201621426678026000997, 9.623088749135040985588830043247, 10.43580575877475140539016390978, 11.74018463814146192703894625092

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