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Properties

Label 2-820-820.427-c0-0-0
Degree $2$
Conductor $820$
Sign $0.808 - 0.588i$
Analytic cond. $0.409233$
Root an. cond. $0.639713$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.987 − 0.156i)5-s + (0.951 + 0.309i)8-s + (0.707 + 0.707i)9-s + (−0.453 + 0.891i)10-s + (−0.678 − 1.10i)13-s + (−0.809 + 0.587i)16-s + (−0.0819 − 1.04i)17-s + (−0.987 + 0.156i)18-s + (−0.453 − 0.891i)20-s + (0.951 − 0.309i)25-s + (1.29 + 0.101i)26-s + (1.20 + 1.40i)29-s − i·32-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.987 − 0.156i)5-s + (0.951 + 0.309i)8-s + (0.707 + 0.707i)9-s + (−0.453 + 0.891i)10-s + (−0.678 − 1.10i)13-s + (−0.809 + 0.587i)16-s + (−0.0819 − 1.04i)17-s + (−0.987 + 0.156i)18-s + (−0.453 − 0.891i)20-s + (0.951 − 0.309i)25-s + (1.29 + 0.101i)26-s + (1.20 + 1.40i)29-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.808 - 0.588i$
Analytic conductor: \(0.409233\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :0),\ 0.808 - 0.588i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8671453404\)
\(L(\frac12)\) \(\approx\) \(0.8671453404\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (-0.987 + 0.156i)T \)
41 \( 1 + (-0.587 + 0.809i)T \)
good3 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (0.891 - 0.453i)T^{2} \)
11 \( 1 + (0.987 - 0.156i)T^{2} \)
13 \( 1 + (0.678 + 1.10i)T + (-0.453 + 0.891i)T^{2} \)
17 \( 1 + (0.0819 + 1.04i)T + (-0.987 + 0.156i)T^{2} \)
19 \( 1 + (-0.891 + 0.453i)T^{2} \)
23 \( 1 + (0.951 - 0.309i)T^{2} \)
29 \( 1 + (-1.20 - 1.40i)T + (-0.156 + 0.987i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.863 - 1.69i)T + (-0.587 - 0.809i)T^{2} \)
43 \( 1 + (-0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.891 - 0.453i)T^{2} \)
53 \( 1 + (0.465 + 0.0366i)T + (0.987 + 0.156i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \)
67 \( 1 + (0.156 - 0.987i)T^{2} \)
71 \( 1 + (-0.987 + 0.156i)T^{2} \)
73 \( 1 - 0.907T + T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.399 + 0.652i)T + (-0.453 - 0.891i)T^{2} \)
97 \( 1 + (0.119 - 0.101i)T + (0.156 - 0.987i)T^{2} \)
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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

−10.27591288787188828522013084840, −9.690908139630296515631076637349, −8.839120773272299379460621932929, −7.917081391272422557060706612905, −7.12475552423512915669237232043, −6.31865699368584511307769335686, −5.10099728204803760986442497556, −4.88763788653604184673712455319, −2.77834122154358238460902753998, −1.41154215384679905759313621866, 1.52544442603589546066140491231, 2.48073275708322022828667356033, 3.83186643998137548233514101405, 4.71604580129303304343560418476, 6.20627194123406475519795023605, 6.91477994955137760544026012038, 7.967145763031785742951063050024, 9.071113955437361945310860163427, 9.552419666700174264377239777849, 10.23543337787606800186779564060

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