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Properties

Label 2-370-185.104-c1-0-16
Degree $2$
Conductor $370$
Sign $0.0249 + 0.999i$
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 + (0.981 − 2.69i)3-s + (0.766 − 0.642i)4-s + (1.64 + 1.51i)5-s − 2.86i·6-s + (−2.07 + 0.365i)7-s + (0.500 − 0.866i)8-s + (−4.00 − 3.36i)9-s + (2.06 + 0.855i)10-s + (2.94 − 5.09i)11-s + (−0.981 − 2.69i)12-s + (−4.66 + 3.91i)13-s + (−1.82 + 1.05i)14-s + (5.69 − 2.96i)15-s + (0.173 − 0.984i)16-s + (5.11 + 4.29i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.566 − 1.55i)3-s + (0.383 − 0.321i)4-s + (0.737 + 0.675i)5-s − 1.17i·6-s + (−0.783 + 0.138i)7-s + (0.176 − 0.306i)8-s + (−1.33 − 1.12i)9-s + (0.653 + 0.270i)10-s + (0.887 − 1.53i)11-s + (−0.283 − 0.778i)12-s + (−1.29 + 1.08i)13-s + (−0.486 + 0.281i)14-s + (1.46 − 0.764i)15-s + (0.0434 − 0.246i)16-s + (1.24 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0249 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0249 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.0249 + 0.999i$
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: \((2,\ 370,\ (\ :1/2),\ 0.0249 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70509 - 1.66314i\)
\(L(\frac12)\) \(\approx\) \(1.70509 - 1.66314i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 1.51i)T \)
37 \( 1 + (2.51 + 5.53i)T \)
good3 \( 1 + (-0.981 + 2.69i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (2.07 - 0.365i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-2.94 + 5.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.66 - 3.91i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-5.11 - 4.29i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (0.963 - 2.64i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (1.71 + 2.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.14 - 1.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.15iT - 31T^{2} \)
41 \( 1 + (1.23 - 1.03i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 1.64T + 43T^{2} \)
47 \( 1 + (8.65 - 4.99i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.41 - 1.65i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (6.20 + 1.09i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-5.62 - 6.70i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.57 + 0.631i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-7.82 - 2.84i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 0.888iT - 73T^{2} \)
79 \( 1 + (-3.75 + 0.662i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.0733 - 0.0874i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-0.148 - 0.0262i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (9.05 + 15.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.53909479766153268623064709005, −10.31370222226354655850964347373, −9.322422735937822395343173588984, −8.250410538324484745583851693493, −7.03730240553716682917742954346, −6.39855842203636059982847490034, −5.78374587660073590807158481716, −3.61998422425026418370649511744, −2.69000863433260554598396507951, −1.54427709120513073258508625661, 2.54302479727314081788245570718, 3.66386614374928683264328475140, 4.85013748809320299207118782366, 5.24130050086556965156695546014, 6.75265547921195075961309653829, 7.933047097360851106001712558794, 9.295693331605049940750534066830, 9.827063540826971073015024473743, 10.16320701298363074007944558482, 11.84933849783178193412427689509

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