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Properties

Label 2-820-820.583-c1-0-11
Degree $2$
Conductor $820$
Sign $-0.997 - 0.0738i$
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  + (−1.39 − 0.219i)2-s − 2.16·3-s + (1.90 + 0.612i)4-s + (−0.498 + 2.17i)5-s + (3.02 + 0.474i)6-s + 0.295i·7-s + (−2.52 − 1.27i)8-s + 1.67·9-s + (1.17 − 2.93i)10-s + (0.592 − 0.592i)11-s + (−4.11 − 1.32i)12-s + 5.60i·13-s + (0.0646 − 0.412i)14-s + (1.07 − 4.71i)15-s + (3.24 + 2.33i)16-s + 5.37i·17-s + ⋯
L(s)  = 1  + (−0.987 − 0.155i)2-s − 1.24·3-s + (0.951 + 0.306i)4-s + (−0.222 + 0.974i)5-s + (1.23 + 0.193i)6-s + 0.111i·7-s + (−0.892 − 0.450i)8-s + 0.559·9-s + (0.371 − 0.928i)10-s + (0.178 − 0.178i)11-s + (−1.18 − 0.382i)12-s + 1.55i·13-s + (0.0172 − 0.110i)14-s + (0.278 − 1.21i)15-s + (0.812 + 0.583i)16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.997 - 0.0738i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.997 - 0.0738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00943025 + 0.255157i\)
\(L(\frac12)\) \(\approx\) \(0.00943025 + 0.255157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.219i)T \)
5 \( 1 + (0.498 - 2.17i)T \)
41 \( 1 + (4.98 - 4.01i)T \)
good3 \( 1 + 2.16T + 3T^{2} \)
7 \( 1 - 0.295iT - 7T^{2} \)
11 \( 1 + (-0.592 + 0.592i)T - 11iT^{2} \)
13 \( 1 - 5.60iT - 13T^{2} \)
17 \( 1 - 5.37iT - 17T^{2} \)
19 \( 1 + (-2.97 + 2.97i)T - 19iT^{2} \)
23 \( 1 + (0.212 - 0.212i)T - 23iT^{2} \)
29 \( 1 + (-2.66 + 2.66i)T - 29iT^{2} \)
31 \( 1 - 4.70iT - 31T^{2} \)
37 \( 1 + (-2.69 + 2.69i)T - 37iT^{2} \)
43 \( 1 + (2.01 + 2.01i)T + 43iT^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + 3.29iT - 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 - 2.98iT - 61T^{2} \)
67 \( 1 + 3.44T + 67T^{2} \)
71 \( 1 + (5.63 - 5.63i)T - 71iT^{2} \)
73 \( 1 + (3.86 - 3.86i)T - 73iT^{2} \)
79 \( 1 + (1.35 + 1.35i)T + 79iT^{2} \)
83 \( 1 + (1.25 - 1.25i)T - 83iT^{2} \)
89 \( 1 + (10.3 - 10.3i)T - 89iT^{2} \)
97 \( 1 + 13.1iT - 97T^{2} \)
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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

−10.67193137322030233692920721129, −10.03689138188313357954532317098, −9.057408327523144791989019339423, −8.120344828701477081739515351723, −6.95201490173580746304357273882, −6.56951053223225026350572462059, −5.76065918528379224030927539461, −4.29558181305881125640660463301, −3.02044759557184463660362322247, −1.59919632278973775952270700566, 0.23234488229203633851864978535, 1.22760635513097762970524687740, 3.09032343198538322451110587136, 4.83827706381675798673708838278, 5.48839395170483131388993294563, 6.26818490101102342037734639147, 7.42239755379128799487085858822, 8.041766890853484866213094280990, 9.039102981581381854096580745817, 9.901107230171838488041535387965

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