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Properties

Label 2-820-41.10-c1-0-8
Degree $2$
Conductor $820$
Sign $-0.789 + 0.614i$
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  − 2.30·3-s + (0.309 + 0.951i)5-s + (0.494 − 0.359i)7-s + 2.33·9-s + (−1.06 + 3.26i)11-s + (−2.36 − 1.71i)13-s + (−0.713 − 2.19i)15-s + (1.17 − 3.61i)17-s + (−1.99 + 1.45i)19-s + (−1.14 + 0.830i)21-s + (5.16 + 3.75i)23-s + (−0.809 + 0.587i)25-s + 1.53·27-s + (−3.23 − 9.96i)29-s + (−2.41 + 7.44i)31-s + ⋯
L(s)  = 1  − 1.33·3-s + (0.138 + 0.425i)5-s + (0.187 − 0.135i)7-s + 0.778·9-s + (−0.319 + 0.984i)11-s + (−0.655 − 0.476i)13-s + (−0.184 − 0.567i)15-s + (0.284 − 0.876i)17-s + (−0.457 + 0.332i)19-s + (−0.249 + 0.181i)21-s + (1.07 + 0.782i)23-s + (−0.161 + 0.117i)25-s + 0.295·27-s + (−0.601 − 1.85i)29-s + (−0.434 + 1.33i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.789 + 0.614i$
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: \((2,\ 820,\ (\ :1/2),\ -0.789 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0574208 - 0.167288i\)
\(L(\frac12)\) \(\approx\) \(0.0574208 - 0.167288i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.40 - 0.00775i)T \)
good3 \( 1 + 2.30T + 3T^{2} \)
7 \( 1 + (-0.494 + 0.359i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.06 - 3.26i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.36 + 1.71i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.17 + 3.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.99 - 1.45i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.16 - 3.75i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (3.23 + 9.96i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.41 - 7.44i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.42 + 7.47i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (9.59 + 6.96i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (10.7 + 7.81i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.74 + 5.35i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.95 + 7.23i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.12 - 0.815i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.483 + 1.48i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.29 + 3.98i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 - 8.61T + 73T^{2} \)
79 \( 1 - 5.67T + 79T^{2} \)
83 \( 1 - 2.05T + 83T^{2} \)
89 \( 1 + (-10.5 + 7.69i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.75 - 11.5i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.11782164540402477401859351896, −9.355738971722503373528935700181, −7.933606692386651582229230450325, −7.13457160698666886146732266724, −6.45273728523146476122583278466, −5.19637618324518579448470644953, −4.99690337158566413383699732960, −3.42199140020649236844832825016, −1.94588847786402346531585976227, −0.10432771344686837424385474737, 1.46082541699292922384039628480, 3.16657045113181463122719731021, 4.69792348102254215582552451717, 5.19773372563285215526978536886, 6.17184714467901731078257240637, 6.81052149496837647198795047725, 8.100623385394508403521105320295, 8.846868211369853003661771610791, 9.882362872264513481013795969807, 10.84758767631891434678739422592

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