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Properties

Label 2-820-20.7-c1-0-39
Degree $2$
Conductor $820$
Sign $-0.584 - 0.811i$
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  + (−1.12 + 0.850i)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 + (1.01 + 2.64i)8-s + 3.01i·9-s + (−1.71 + 2.65i)10-s + 4.02i·11-s + (4.29 − 2.37i)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  + (−0.798 + 0.601i)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 + (0.357 + 0.933i)8-s + 1.00i·9-s + (−0.543 + 0.839i)10-s + 1.21i·11-s + (1.23 − 0.685i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.584 - 0.811i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.584 - 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.708882 + 1.38404i\)
\(L(\frac12)\) \(\approx\) \(0.708882 + 1.38404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 - 0.850i)T \)
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

−10.10087761792402229109692195689, −9.509021295798309868955256628185, −9.003844521252919156813543127849, −8.373470924239241490981491289859, −7.15667860705911520436798473268, −6.31246559091472369377795973293, −5.17773993011667377656143513929, −4.42489633430785054914548974189, −2.80076920232169193056671078987, −1.83981367142285065798452332820, 0.891026530746769056787687856113, 2.25657171579917319201556060729, 2.83234307402179150479692215911, 3.95985515618478882272889933273, 5.95248729777663873313011858985, 6.72503480650282377320763538502, 7.57142020802499942975944791417, 8.347977679952194613192162715050, 9.051236713688374192013217572307, 9.818822940993404959075103646151

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