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Properties

Label 2-820-205.159-c1-0-0
Degree $2$
Conductor $820$
Sign $0.262 - 0.965i$
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  + (−1.37 − 1.37i)3-s + (−2.23 − 0.145i)5-s + (−0.0463 − 0.292i)7-s + 0.802i·9-s + (1.40 + 0.714i)11-s + (−3.46 − 0.548i)13-s + (2.87 + 3.27i)15-s + (−3.94 − 2.00i)17-s + (0.467 + 2.94i)19-s + (−0.339 + 0.467i)21-s + (−1.12 − 1.54i)23-s + (4.95 + 0.649i)25-s + (−3.03 + 3.03i)27-s + (4.61 + 9.05i)29-s + (−0.629 − 1.93i)31-s + ⋯
L(s)  = 1  + (−0.796 − 0.796i)3-s + (−0.997 − 0.0650i)5-s + (−0.0175 − 0.110i)7-s + 0.267i·9-s + (0.422 + 0.215i)11-s + (−0.960 − 0.152i)13-s + (0.742 + 0.846i)15-s + (−0.956 − 0.487i)17-s + (0.107 + 0.676i)19-s + (−0.0740 + 0.101i)21-s + (−0.234 − 0.322i)23-s + (0.991 + 0.129i)25-s + (−0.583 + 0.583i)27-s + (0.856 + 1.68i)29-s + (−0.113 − 0.347i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.262 - 0.965i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.262 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296763 + 0.226926i\)
\(L(\frac12)\) \(\approx\) \(0.296763 + 0.226926i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.145i)T \)
41 \( 1 + (-0.449 - 6.38i)T \)
good3 \( 1 + (1.37 + 1.37i)T + 3iT^{2} \)
7 \( 1 + (0.0463 + 0.292i)T + (-6.65 + 2.16i)T^{2} \)
11 \( 1 + (-1.40 - 0.714i)T + (6.46 + 8.89i)T^{2} \)
13 \( 1 + (3.46 + 0.548i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (3.94 + 2.00i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.467 - 2.94i)T + (-18.0 + 5.87i)T^{2} \)
23 \( 1 + (1.12 + 1.54i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-4.61 - 9.05i)T + (-17.0 + 23.4i)T^{2} \)
31 \( 1 + (0.629 + 1.93i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-9.17 - 2.98i)T + (29.9 + 21.7i)T^{2} \)
43 \( 1 + (6.88 - 5.00i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (0.920 - 5.81i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-6.34 + 3.23i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (7.47 - 5.43i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.99 + 5.49i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-6.17 - 12.1i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (11.0 + 5.64i)T + (41.7 + 57.4i)T^{2} \)
73 \( 1 + 7.00T + 73T^{2} \)
79 \( 1 + (1.93 - 1.93i)T - 79iT^{2} \)
83 \( 1 + 6.08iT - 83T^{2} \)
89 \( 1 + (8.22 - 1.30i)T + (84.6 - 27.5i)T^{2} \)
97 \( 1 + (2.14 + 4.20i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.59729009077305637465106020508, −9.585112985480382705530074282249, −8.573653065366210770739072611002, −7.62259575337628759524670577831, −6.97254432083437571812326088944, −6.27657638735373726087728383549, −5.03668369620743783386709812724, −4.21004973217940068082035296195, −2.85656249427210279954207466750, −1.18504565252135379551766118216, 0.23125823506153667716519966031, 2.49227671068392649746736325612, 3.99898928511753519433038318107, 4.49801841694759313745056991164, 5.49286121091213986482089371497, 6.54701080858682771340527790961, 7.44832618122505030264594014232, 8.405673358777475804642034194476, 9.311536908322566320728514092422, 10.22431976609819562223974809595

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