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Properties

Label 2-370-185.174-c1-0-5
Degree $2$
Conductor $370$
Sign $0.985 + 0.168i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.133 + 2.23i)5-s + (0.633 + 0.366i)7-s − 0.999i·8-s + (−1.5 − 2.59i)9-s + (1.23 + 1.86i)10-s + 4.73·11-s + (4.73 + 2.73i)13-s + 0.732·14-s + (−0.5 − 0.866i)16-s + (−1.5 + 0.866i)17-s + (−2.59 − 1.5i)18-s + (2.36 − 4.09i)19-s + (1.99 + 0.999i)20-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.0599 + 0.998i)5-s + (0.239 + 0.138i)7-s − 0.353i·8-s + (−0.5 − 0.866i)9-s + (0.389 + 0.590i)10-s + 1.42·11-s + (1.31 + 0.757i)13-s + 0.195·14-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.210i)17-s + (−0.612 − 0.353i)18-s + (0.542 − 0.940i)19-s + (0.447 + 0.223i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.985 + 0.168i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.985 + 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02029 - 0.171573i\)
\(L(\frac12)\) \(\approx\) \(2.02029 - 0.171573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.133 - 2.23i)T \)
37 \( 1 + (2.59 + 5.5i)T \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.633 - 0.366i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 + (-4.73 - 2.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 - 0.866i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.36 + 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.46iT - 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 + 9.66T + 31T^{2} \)
41 \( 1 + (-3.96 + 6.86i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 5.26iT - 43T^{2} \)
47 \( 1 + 3.26iT - 47T^{2} \)
53 \( 1 + (10.7 - 6.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.26 + 2.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.13 - 3.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.56 - 2.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.56 - 13.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.8iT - 73T^{2} \)
79 \( 1 + (0.830 - 1.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.19 + 4.73i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.232 - 0.401i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.19iT - 97T^{2} \)
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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

−11.32460650056526887569031844738, −10.96091691286465770131936734289, −9.389688063826917547571119040237, −8.991777886711284999933790735884, −7.25155308099583979509767833446, −6.45660061565450139267834914596, −5.69832737446367516151538623705, −3.97058560409520192777649391883, −3.36854791999168132351616504737, −1.70776531494346732372960374905, 1.56430437007422083752390364431, 3.49082784379341259265791973288, 4.54315880618533506643770970511, 5.55479424027320720636749876189, 6.38372774454026419802330215857, 7.81300621648394385768618414519, 8.477843343502342147954019825088, 9.357085194345719055693859439596, 10.78479616242656875812711085857, 11.50472665250096951937272538187

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