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Properties

Label 2-820-20.3-c1-0-36
Degree $2$
Conductor $820$
Sign $-0.926 + 0.377i$
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  + (0.850 + 1.12i)2-s + (−1.73 + 1.73i)3-s + (−0.552 + 1.92i)4-s + (2.10 + 0.767i)5-s + (−3.43 − 0.484i)6-s + (1.19 + 1.19i)7-s + (−2.64 + 1.01i)8-s − 3.01i·9-s + (0.918 + 3.02i)10-s + 4.02i·11-s + (−2.37 − 4.29i)12-s + (−0.809 − 0.809i)13-s + (−0.332 + 2.36i)14-s + (−4.97 + 2.31i)15-s + (−3.38 − 2.12i)16-s + (−0.339 + 0.339i)17-s + ⋯
L(s)  = 1  + (0.601 + 0.798i)2-s + (−1.00 + 1.00i)3-s + (−0.276 + 0.961i)4-s + (0.939 + 0.343i)5-s + (−1.40 − 0.197i)6-s + (0.450 + 0.450i)7-s + (−0.933 + 0.357i)8-s − 1.00i·9-s + (0.290 + 0.956i)10-s + 1.21i·11-s + (−0.685 − 1.23i)12-s + (−0.224 − 0.224i)13-s + (−0.0889 + 0.631i)14-s + (−1.28 + 0.596i)15-s + (−0.847 − 0.531i)16-s + (−0.0823 + 0.0823i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.926 + 0.377i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.926 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.310152 - 1.58433i\)
\(L(\frac12)\) \(\approx\) \(0.310152 - 1.58433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.850 - 1.12i)T \)
5 \( 1 + (-2.10 - 0.767i)T \)
41 \( 1 - T \)
good3 \( 1 + (1.73 - 1.73i)T - 3iT^{2} \)
7 \( 1 + (-1.19 - 1.19i)T + 7iT^{2} \)
11 \( 1 - 4.02iT - 11T^{2} \)
13 \( 1 + (0.809 + 0.809i)T + 13iT^{2} \)
17 \( 1 + (0.339 - 0.339i)T - 17iT^{2} \)
19 \( 1 - 4.20T + 19T^{2} \)
23 \( 1 + (1.29 - 1.29i)T - 23iT^{2} \)
29 \( 1 + 4.18iT - 29T^{2} \)
31 \( 1 - 5.69iT - 31T^{2} \)
37 \( 1 + (-4.60 + 4.60i)T - 37iT^{2} \)
43 \( 1 + (-0.412 + 0.412i)T - 43iT^{2} \)
47 \( 1 + (8.15 + 8.15i)T + 47iT^{2} \)
53 \( 1 + (-0.269 - 0.269i)T + 53iT^{2} \)
59 \( 1 - 5.44T + 59T^{2} \)
61 \( 1 - 0.729T + 61T^{2} \)
67 \( 1 + (9.31 + 9.31i)T + 67iT^{2} \)
71 \( 1 - 1.57iT - 71T^{2} \)
73 \( 1 + (-5.88 - 5.88i)T + 73iT^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 + (-3.59 + 3.59i)T - 83iT^{2} \)
89 \( 1 - 6.99iT - 89T^{2} \)
97 \( 1 + (9.75 - 9.75i)T - 97iT^{2} \)
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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

−10.64826341989030520089551957210, −9.793347585408345840697807802440, −9.276453182582220434284895000932, −7.979745520290617164092928376345, −6.98000853516536213242316242978, −6.11502458654967449495801670392, −5.22854556940186782768306729850, −4.94197184378354857677661283963, −3.73956410831653032883792576968, −2.28510781322098207478916255164, 0.77736942945106426674362120361, 1.61966283799048386503029633588, 2.96402925899104675959527767263, 4.49001990479266037950876109020, 5.43936314189062766315084270849, 6.00919536329552849536565010011, 6.77561470763104831811506512739, 7.991146745935633392151784307214, 9.166517236759108335720997587010, 9.977915112030912225535731147831

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