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Properties

Label 2-820-820.643-c0-0-0
Degree $2$
Conductor $820$
Sign $-0.700 - 0.713i$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.453 + 0.891i)5-s + (−0.587 + 0.809i)8-s + (−0.707 + 0.707i)9-s + (0.156 − 0.987i)10-s + (−0.355 + 0.303i)13-s + (0.309 − 0.951i)16-s + (−1.29 + 0.794i)17-s + (0.453 − 0.891i)18-s + (0.156 + 0.987i)20-s + (−0.587 − 0.809i)25-s + (0.243 − 0.398i)26-s + (0.178 + 0.744i)29-s + i·32-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−0.453 + 0.891i)5-s + (−0.587 + 0.809i)8-s + (−0.707 + 0.707i)9-s + (0.156 − 0.987i)10-s + (−0.355 + 0.303i)13-s + (0.309 − 0.951i)16-s + (−1.29 + 0.794i)17-s + (0.453 − 0.891i)18-s + (0.156 + 0.987i)20-s + (−0.587 − 0.809i)25-s + (0.243 − 0.398i)26-s + (0.178 + 0.744i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.700 - 0.713i$
Analytic conductor: \(0.409233\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :0),\ -0.700 - 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4027266970\)
\(L(\frac12)\) \(\approx\) \(0.4027266970\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 + (0.453 - 0.891i)T \)
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 \( 1 + (0.355 - 0.303i)T + (0.156 - 0.987i)T^{2} \)
17 \( 1 + (1.29 - 0.794i)T + (0.453 - 0.891i)T^{2} \)
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 \( 1 + (0.183 - 1.16i)T + (-0.951 - 0.309i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (-0.987 - 0.156i)T^{2} \)
53 \( 1 + (0.0819 - 0.133i)T + (-0.453 - 0.891i)T^{2} \)
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 \( 1 + 0.312T + T^{2} \)
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 \( 1 + (-1.65 + 0.398i)T + (0.891 - 0.453i)T^{2} \)
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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

−10.71575663096101590012977431611, −10.00413352145978645837839345223, −8.870422418042184164442511411292, −8.271835496884370613049443740886, −7.38289312071140798477759221708, −6.65521737418858050582876598916, −5.81746083515091577478259866575, −4.54732721342845712843693906207, −3.02408178600221431256045396979, −2.04988732026400040769542014670, 0.52630451104177643059474481458, 2.26198746414356185445250173190, 3.47172129686705503810237623599, 4.61324391963298476366394941671, 5.86971913920598927665399182954, 6.88467274200435391719714217203, 7.81447093990793772339087620188, 8.596100703075328495838275688927, 9.217803458260765221855356852427, 9.860500103249012817403534933395

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