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Properties

Label 2-820-205.137-c1-0-19
Degree $2$
Conductor $820$
Sign $-0.917 + 0.397i$
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  + (1.03 + 0.430i)3-s + (−2.16 − 0.554i)5-s + (−0.831 + 2.00i)7-s + (−1.22 − 1.22i)9-s + (2.50 − 1.03i)11-s + (−5.26 − 2.17i)13-s + (−2.01 − 1.50i)15-s + (−5.37 + 2.22i)17-s + (1.89 + 0.786i)19-s + (−1.72 + 1.72i)21-s + (−3.33 − 3.33i)23-s + (4.38 + 2.40i)25-s + (−2.03 − 4.91i)27-s + (−3.07 + 1.27i)29-s − 0.621i·31-s + ⋯
L(s)  = 1  + (0.599 + 0.248i)3-s + (−0.968 − 0.248i)5-s + (−0.314 + 0.758i)7-s + (−0.409 − 0.409i)9-s + (0.755 − 0.313i)11-s + (−1.45 − 0.604i)13-s + (−0.519 − 0.389i)15-s + (−1.30 + 0.540i)17-s + (0.435 + 0.180i)19-s + (−0.377 + 0.377i)21-s + (−0.694 − 0.694i)23-s + (0.876 + 0.480i)25-s + (−0.392 − 0.946i)27-s + (−0.571 + 0.236i)29-s − 0.111i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.917 + 0.397i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.917 + 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0395447 - 0.190796i\)
\(L(\frac12)\) \(\approx\) \(0.0395447 - 0.190796i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 + 0.554i)T \)
41 \( 1 + (6.40 - 0.196i)T \)
good3 \( 1 + (-1.03 - 0.430i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (0.831 - 2.00i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.50 + 1.03i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (5.26 + 2.17i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + (5.37 - 2.22i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (-1.89 - 0.786i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.33 + 3.33i)T + 23iT^{2} \)
29 \( 1 + (3.07 - 1.27i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 0.621iT - 31T^{2} \)
37 \( 1 + (5.98 + 5.98i)T + 37iT^{2} \)
43 \( 1 - 1.18T + 43T^{2} \)
47 \( 1 + (2.97 - 1.23i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (-3.00 + 7.25i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 - 8.81iT - 59T^{2} \)
61 \( 1 + (-1.18 - 1.18i)T + 61iT^{2} \)
67 \( 1 + (4.71 - 1.95i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (3.30 + 7.97i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 - 7.49T + 73T^{2} \)
79 \( 1 + (-2.70 - 6.54i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (3.55 - 3.55i)T - 83iT^{2} \)
89 \( 1 + (-5.23 + 2.16i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (0.940 + 2.27i)T + (-68.5 + 68.5i)T^{2} \)
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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

−9.637605517730107133499736513665, −8.889444819553825143632745008409, −8.415324277528329082110776046094, −7.41076928599808599509110925362, −6.42878288483618921671134651101, −5.35078127428024159396532324309, −4.21347098895406609824341891509, −3.36952555049543642139357780695, −2.32323577209898171154418334089, −0.082392200469765516571056461442, 2.02540242851761837271923292741, 3.19661597549285870150062143397, 4.17450916145025525669769136964, 5.03868982343184128289544939941, 6.74098547252654816577796126200, 7.17308979518718962425332000986, 7.912934887211003157620814935427, 8.900878188460035518401617326329, 9.643149450821215686758794556363, 10.60061657909504677647313564239

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