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Properties

Label 2-370-185.84-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.504 - 0.863i$
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 + (1.86 + 1.23i)5-s + (−3.46 + 2i)7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (−1 − 2i)10-s − 3·11-s + (−0.866 + 0.5i)13-s + 3.99·14-s + (−0.5 + 0.866i)16-s + (−5.19 − 3i)17-s + (2.59 − 1.5i)18-s + (−1.5 − 2.59i)19-s + (−0.133 + 2.23i)20-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.834 + 0.550i)5-s + (−1.30 + 0.755i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.316 − 0.632i)10-s − 0.904·11-s + (−0.240 + 0.138i)13-s + 1.06·14-s + (−0.125 + 0.216i)16-s + (−1.26 − 0.727i)17-s + (0.612 − 0.353i)18-s + (−0.344 − 0.596i)19-s + (−0.0299 + 0.499i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.504 - 0.863i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.504 - 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267966 + 0.466746i\)
\(L(\frac12)\) \(\approx\) \(0.267966 + 0.466746i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 - 1.23i)T \)
37 \( 1 + (-2.59 - 5.5i)T \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (3.46 - 2i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
41 \( 1 + (-5 - 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 - 11iT - 47T^{2} \)
53 \( 1 + (-8.66 - 5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.19 - 3i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−11.37028410495232689632975315207, −10.75760950011147263754865559787, −9.754855510633295679946028150713, −9.231886045792311147283172035205, −8.154707599900620657938405971372, −6.90944550808177087228869834061, −6.12911843667100302064604015565, −4.93361890103380899736607386423, −2.81772696693861262975472687071, −2.47065844549861953470922904189, 0.40222237310878924378328645053, 2.42756932472951083495926522505, 3.98976550893068370746307652244, 5.58251888620118908566387145747, 6.33406710812973675245333743579, 7.18836893378786406348373422968, 8.594525999363965002163324805060, 9.141238109604676314721867358454, 10.24257850079152571235476380668, 10.52285556739154514188875860883

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