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Properties

Label 2-820-41.16-c1-0-4
Degree $2$
Conductor $820$
Sign $0.445 - 0.895i$
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  − 0.355·3-s + (0.809 + 0.587i)5-s + (0.307 + 0.946i)7-s − 2.87·9-s + (3.39 − 2.46i)11-s + (−1.59 + 4.90i)13-s + (−0.287 − 0.209i)15-s + (2.28 − 1.65i)17-s + (2.17 + 6.68i)19-s + (−0.109 − 0.336i)21-s + (−0.101 + 0.312i)23-s + (0.309 + 0.951i)25-s + 2.08·27-s + (3.80 + 2.76i)29-s + (−2.25 + 1.64i)31-s + ⋯
L(s)  = 1  − 0.205·3-s + (0.361 + 0.262i)5-s + (0.116 + 0.357i)7-s − 0.957·9-s + (1.02 − 0.743i)11-s + (−0.441 + 1.36i)13-s + (−0.0743 − 0.0539i)15-s + (0.554 − 0.402i)17-s + (0.498 + 1.53i)19-s + (−0.0238 − 0.0734i)21-s + (−0.0211 + 0.0650i)23-s + (0.0618 + 0.190i)25-s + 0.402·27-s + (0.706 + 0.513i)29-s + (−0.405 + 0.294i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.445 - 0.895i$
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: \((2,\ 820,\ (\ :1/2),\ 0.445 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20153 + 0.743788i\)
\(L(\frac12)\) \(\approx\) \(1.20153 + 0.743788i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.39 - 0.316i)T \)
good3 \( 1 + 0.355T + 3T^{2} \)
7 \( 1 + (-0.307 - 0.946i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-3.39 + 2.46i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.59 - 4.90i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.28 + 1.65i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.17 - 6.68i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.101 - 0.312i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-3.80 - 2.76i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.25 - 1.64i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (6.30 + 4.58i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (3.73 - 11.5i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.76 - 5.42i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-8.16 - 5.93i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.06 - 3.26i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.03 + 12.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-7.67 - 5.57i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (11.8 - 8.61i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 - 6.31T + 73T^{2} \)
79 \( 1 - 8.58T + 79T^{2} \)
83 \( 1 - 3.73T + 83T^{2} \)
89 \( 1 + (4.09 + 12.6i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-9.84 - 7.15i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.39590109946564389416420378382, −9.385431542632629806517873171844, −8.883971413567279907960461003490, −7.86908188225881239399784324704, −6.74944567289250589666392075471, −6.00922918902415183431887354553, −5.25196255558179139899872511910, −3.92316085178557493272748577180, −2.87139173206002953889251911530, −1.48925904307487280676240204215, 0.77156939091935421667549618268, 2.40666611473640777100803480576, 3.59531248090989077778697221685, 4.88587895638669273974768871043, 5.55484885463505584234316757573, 6.61217219345415520425449476967, 7.47299624240237237674393052576, 8.476476956502374635349165803862, 9.237271486929839322170453935313, 10.13567450582259558150059799397

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