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Properties

Label 2-820-41.37-c1-0-10
Degree $2$
Conductor $820$
Sign $0.0877 + 0.996i$
Analytic cond. $6.54773$
Root an. cond. $2.55885$
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  + 0.0908·3-s + (−0.309 + 0.951i)5-s + (−1.97 − 1.43i)7-s − 2.99·9-s + (0.753 + 2.31i)11-s + (4.00 − 2.91i)13-s + (−0.0280 + 0.0864i)15-s + (−2.34 − 7.22i)17-s + (0.0424 + 0.0308i)19-s + (−0.179 − 0.130i)21-s + (7.24 − 5.26i)23-s + (−0.809 − 0.587i)25-s − 0.544·27-s + (0.740 − 2.27i)29-s + (−2.43 − 7.50i)31-s + ⋯
L(s)  = 1  + 0.0524·3-s + (−0.138 + 0.425i)5-s + (−0.747 − 0.542i)7-s − 0.997·9-s + (0.227 + 0.699i)11-s + (1.11 − 0.807i)13-s + (−0.00724 + 0.0223i)15-s + (−0.569 − 1.75i)17-s + (0.00973 + 0.00707i)19-s + (−0.0391 − 0.0284i)21-s + (1.51 − 1.09i)23-s + (−0.161 − 0.117i)25-s − 0.104·27-s + (0.137 − 0.423i)29-s + (−0.437 − 1.34i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0877 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0877 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.0877 + 0.996i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.0877 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762761 - 0.698519i\)
\(L(\frac12)\) \(\approx\) \(0.762761 - 0.698519i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (4.35 + 4.69i)T \)
good3 \( 1 - 0.0908T + 3T^{2} \)
7 \( 1 + (1.97 + 1.43i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.753 - 2.31i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-4.00 + 2.91i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.34 + 7.22i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.0424 - 0.0308i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-7.24 + 5.26i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.740 + 2.27i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.43 + 7.50i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.18 - 6.73i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (0.252 - 0.183i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-6.04 + 4.38i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.80 - 11.7i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.95 + 5.05i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.37 + 3.90i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.08 - 6.42i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.546 - 1.68i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 5.06T + 73T^{2} \)
79 \( 1 + 4.21T + 79T^{2} \)
83 \( 1 - 7.09T + 83T^{2} \)
89 \( 1 + (4.25 + 3.09i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.55 + 7.86i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.06103552052001682464219137085, −9.182624354272515702237150358985, −8.427544580270414219522633016969, −7.28286686510676886513350832493, −6.68990588941667827794644427409, −5.70899099260001806381672570129, −4.57618344448717394280609671666, −3.36938159859473785957969086960, −2.61508456027465797888624059799, −0.52029768424503162852316594258, 1.51670277294584929124269448159, 3.12181113106342196921489334884, 3.84370920591379568652423434698, 5.29135846808633301209043421163, 6.06912665822373330579148030936, 6.76411147465609767819807588821, 8.181847898581023069806206195889, 8.991732437177618487400954803293, 9.066610522388176267416377535528, 10.65812392917614515530467848272

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