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Properties

Label 2-820-41.10-c1-0-1
Degree $2$
Conductor $820$
Sign $-0.794 - 0.607i$
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  + 1.59·3-s + (−0.309 − 0.951i)5-s + (−3.74 + 2.72i)7-s − 0.452·9-s + (−1.85 + 5.70i)11-s + (−2.33 − 1.69i)13-s + (−0.493 − 1.51i)15-s + (1.55 − 4.79i)17-s + (−6.33 + 4.59i)19-s + (−5.98 + 4.34i)21-s + (3.04 + 2.21i)23-s + (−0.809 + 0.587i)25-s − 5.51·27-s + (−2.33 − 7.19i)29-s + (0.617 − 1.90i)31-s + ⋯
L(s)  = 1  + 0.921·3-s + (−0.138 − 0.425i)5-s + (−1.41 + 1.02i)7-s − 0.150·9-s + (−0.558 + 1.71i)11-s + (−0.647 − 0.470i)13-s + (−0.127 − 0.391i)15-s + (0.377 − 1.16i)17-s + (−1.45 + 1.05i)19-s + (−1.30 + 0.948i)21-s + (0.634 + 0.460i)23-s + (−0.161 + 0.117i)25-s − 1.06·27-s + (−0.434 − 1.33i)29-s + (0.110 − 0.341i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.794 - 0.607i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.219338 + 0.647406i\)
\(L(\frac12)\) \(\approx\) \(0.219338 + 0.647406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-1.63 - 6.19i)T \)
good3 \( 1 - 1.59T + 3T^{2} \)
7 \( 1 + (3.74 - 2.72i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.85 - 5.70i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.33 + 1.69i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.55 + 4.79i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.33 - 4.59i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-3.04 - 2.21i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.33 + 7.19i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.617 + 1.90i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.05 - 3.24i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-2.56 - 1.86i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-1.08 - 0.789i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.44 - 10.6i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.30 - 3.12i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.56 + 1.13i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.19 - 3.67i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.79 + 11.6i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + (10.1 - 7.39i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.03 - 9.32i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.04601963921996434919543019562, −9.676957837072556860728694843701, −9.031354362140588183854436534983, −8.027965920123031963833758963150, −7.36782808528064397984907446533, −6.18797376915512431935101038373, −5.26975947201670102050162828671, −4.12055365384704927552286688838, −2.82578060538178861525999863647, −2.30922383306505909861089541740, 0.27097081611025724027963585614, 2.51557659845713040803651665514, 3.33563527387440087851366879287, 3.98453870868560102275611502488, 5.62170612328965550125651324850, 6.62323070211346416631110767489, 7.23237880130361391759914656984, 8.449089250397376413207068768478, 8.836159068709762507445778343263, 9.923672441424313729000334114985

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