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Properties

Label 2-370-185.68-c1-0-9
Degree $2$
Conductor $370$
Sign $0.458 + 0.888i$
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  − 2-s + (0.536 − 0.536i)3-s + 4-s + (−2.23 + 0.127i)5-s + (−0.536 + 0.536i)6-s + (0.767 − 0.767i)7-s − 8-s + 2.42i·9-s + (2.23 − 0.127i)10-s − 4.39i·11-s + (0.536 − 0.536i)12-s + 6.74·13-s + (−0.767 + 0.767i)14-s + (−1.12 + 1.26i)15-s + 16-s − 7.34i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.309 − 0.309i)3-s + 0.5·4-s + (−0.998 + 0.0571i)5-s + (−0.219 + 0.219i)6-s + (0.290 − 0.290i)7-s − 0.353·8-s + 0.808i·9-s + (0.705 − 0.0404i)10-s − 1.32i·11-s + (0.154 − 0.154i)12-s + 1.87·13-s + (−0.205 + 0.205i)14-s + (−0.291 + 0.326i)15-s + 0.250·16-s − 1.78i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.458 + 0.888i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.458 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802326 - 0.489132i\)
\(L(\frac12)\) \(\approx\) \(0.802326 - 0.489132i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (2.23 - 0.127i)T \)
37 \( 1 + (-6.04 + 0.633i)T \)
good3 \( 1 + (-0.536 + 0.536i)T - 3iT^{2} \)
7 \( 1 + (-0.767 + 0.767i)T - 7iT^{2} \)
11 \( 1 + 4.39iT - 11T^{2} \)
13 \( 1 - 6.74T + 13T^{2} \)
17 \( 1 + 7.34iT - 17T^{2} \)
19 \( 1 + (2.59 + 2.59i)T + 19iT^{2} \)
23 \( 1 - 1.20T + 23T^{2} \)
29 \( 1 + (1.25 - 1.25i)T - 29iT^{2} \)
31 \( 1 + (4.14 + 4.14i)T + 31iT^{2} \)
41 \( 1 + 4.07iT - 41T^{2} \)
43 \( 1 + 8.56T + 43T^{2} \)
47 \( 1 + (-7.68 + 7.68i)T - 47iT^{2} \)
53 \( 1 + (-2.31 - 2.31i)T + 53iT^{2} \)
59 \( 1 + (-7.61 - 7.61i)T + 59iT^{2} \)
61 \( 1 + (-1.14 - 1.14i)T + 61iT^{2} \)
67 \( 1 + (-6.25 - 6.25i)T + 67iT^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + (1.88 - 1.88i)T - 73iT^{2} \)
79 \( 1 + (5.13 + 5.13i)T + 79iT^{2} \)
83 \( 1 + (-0.570 - 0.570i)T + 83iT^{2} \)
89 \( 1 + (7.54 - 7.54i)T - 89iT^{2} \)
97 \( 1 - 5.39iT - 97T^{2} \)
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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

−11.15942386887266794319881225171, −10.63888344079404118892256147635, −8.976016077829635454927211228455, −8.497724961830795629062060478467, −7.65547178197472295655974375457, −6.84023544553937522234617642602, −5.48312056699927288382531036098, −3.96180408119657111913556146283, −2.76886333314809316531144963513, −0.865287218671018101826048689564, 1.57498701703147038738978322923, 3.50279409985366115401881304839, 4.25182232808081372223240095544, 6.01441385892366341713050377685, 6.94958425922972055595891682828, 8.307635307273956549335690547932, 8.486896700602345151557878900724, 9.651768416117621174174194272689, 10.63540929523805678505360724096, 11.36556820883056863407576363104

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