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Properties

Label 2-820-41.16-c1-0-1
Degree $2$
Conductor $820$
Sign $-0.641 - 0.766i$
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  + 0.300·3-s + (0.809 + 0.587i)5-s + (−0.378 − 1.16i)7-s − 2.90·9-s + (−4.91 + 3.56i)11-s + (−0.794 + 2.44i)13-s + (0.243 + 0.176i)15-s + (−1.01 + 0.735i)17-s + (0.436 + 1.34i)19-s + (−0.113 − 0.350i)21-s + (−2.05 + 6.33i)23-s + (0.309 + 0.951i)25-s − 1.77·27-s + (−2.33 − 1.69i)29-s + (−0.348 + 0.253i)31-s + ⋯
L(s)  = 1  + 0.173·3-s + (0.361 + 0.262i)5-s + (−0.143 − 0.440i)7-s − 0.969·9-s + (−1.48 + 1.07i)11-s + (−0.220 + 0.677i)13-s + (0.0628 + 0.0456i)15-s + (−0.245 + 0.178i)17-s + (0.100 + 0.308i)19-s + (−0.0248 − 0.0764i)21-s + (−0.429 + 1.32i)23-s + (0.0618 + 0.190i)25-s − 0.342·27-s + (−0.433 − 0.314i)29-s + (−0.0626 + 0.0454i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.641 - 0.766i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.641 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.325680 + 0.697386i\)
\(L(\frac12)\) \(\approx\) \(0.325680 + 0.697386i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-2.84 + 5.73i)T \)
good3 \( 1 - 0.300T + 3T^{2} \)
7 \( 1 + (0.378 + 1.16i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (4.91 - 3.56i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.794 - 2.44i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.01 - 0.735i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.436 - 1.34i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.05 - 6.33i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.33 + 1.69i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.348 - 0.253i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-8.65 - 6.29i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-0.135 + 0.417i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-0.555 + 1.70i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.37 + 5.36i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.68 - 11.3i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.26 - 6.96i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.20 + 5.23i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-4.60 + 3.34i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 1.52T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + (1.28 + 3.95i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.63 - 4.09i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.37826279048979737793726612313, −9.806208776782352770895882307843, −8.934001577604030259940529503645, −7.80241528304664576534654480172, −7.30922219622219448100920462190, −6.10713786259103973953781872712, −5.29601780715002287373248065538, −4.20730939548088243112389197624, −2.92920642295344618252370790234, −1.97629431276445599510901061858, 0.33780351776656018808649036536, 2.50027550445368906795554885413, 3.02882476182727652708343378237, 4.66815818188774017104373864374, 5.67788794913045211105391536603, 6.07553443060540296327488167875, 7.60577717039255067406508913430, 8.307840520598259967465736427763, 8.955104106787172154781032852787, 9.899333075765628123169970960682

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