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Properties

Label 2-820-820.807-c0-0-0
Degree $2$
Conductor $820$
Sign $-0.588 + 0.808i$
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  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.156 − 0.987i)5-s + (0.809 + 0.587i)8-s + (0.707 − 0.707i)9-s + (−0.891 + 0.453i)10-s + (0.303 + 0.355i)13-s + (0.309 − 0.951i)16-s + (−0.794 − 1.29i)17-s + (−0.891 − 0.453i)18-s + (0.707 + 0.707i)20-s + (−0.951 + 0.309i)25-s + (0.243 − 0.398i)26-s + (−0.178 − 0.744i)29-s − 32-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.156 − 0.987i)5-s + (0.809 + 0.587i)8-s + (0.707 − 0.707i)9-s + (−0.891 + 0.453i)10-s + (0.303 + 0.355i)13-s + (0.309 − 0.951i)16-s + (−0.794 − 1.29i)17-s + (−0.891 − 0.453i)18-s + (0.707 + 0.707i)20-s + (−0.951 + 0.309i)25-s + (0.243 − 0.398i)26-s + (−0.178 − 0.744i)29-s − 32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.588 + 0.808i$
Analytic conductor: \(0.409233\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (807, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :0),\ -0.588 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7533746941\)
\(L(\frac12)\) \(\approx\) \(0.7533746941\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.156 + 0.987i)T \)
41 \( 1 + (-0.951 + 0.309i)T \)
good3 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.987 + 0.156i)T^{2} \)
11 \( 1 + (-0.453 + 0.891i)T^{2} \)
13 \( 1 + (-0.303 - 0.355i)T + (-0.156 + 0.987i)T^{2} \)
17 \( 1 + (0.794 + 1.29i)T + (-0.453 + 0.891i)T^{2} \)
19 \( 1 + (-0.987 + 0.156i)T^{2} \)
23 \( 1 + (0.587 + 0.809i)T^{2} \)
29 \( 1 + (0.178 + 0.744i)T + (-0.891 + 0.453i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (1.16 + 0.183i)T + (0.951 + 0.309i)T^{2} \)
43 \( 1 + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (0.987 + 0.156i)T^{2} \)
53 \( 1 + (-0.133 - 0.0819i)T + (0.453 + 0.891i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2} \)
67 \( 1 + (-0.891 + 0.453i)T^{2} \)
71 \( 1 + (0.453 - 0.891i)T^{2} \)
73 \( 1 - 0.312iT - T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-1.40 - 1.20i)T + (0.156 + 0.987i)T^{2} \)
97 \( 1 + (-0.398 - 1.65i)T + (-0.891 + 0.453i)T^{2} \)
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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

−10.04847725696315063832189723517, −9.207133282519870228480371374938, −8.898081437361407517663094047700, −7.78143137877388102754016235971, −6.87750715998059904078306069143, −5.43244084047869630967163521457, −4.43671821658295478706119599252, −3.78217596203016693217327269964, −2.29665218294532170545699391445, −0.935923241922551341564064401155, 1.88860688591616982141893676877, 3.58974654985044532089673322782, 4.55869480102355915479464450710, 5.70647385763529402829556307336, 6.56593341423101306085417750457, 7.27293438044206327081446359982, 8.043863552430213099809131196689, 8.840496157704396668915854029556, 9.976647473503105786229855027139, 10.57324401531560137603409968742

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