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Properties

Label 2-820-41.25-c1-0-7
Degree $2$
Conductor $820$
Sign $0.919 + 0.392i$
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  − 0.195i·3-s + (−0.809 − 0.587i)5-s + (0.384 − 0.124i)7-s + 2.96·9-s + (1.29 + 1.78i)11-s + (−0.0308 − 0.0100i)13-s + (−0.115 + 0.158i)15-s + (−2.32 − 3.19i)17-s + (3.48 − 1.13i)19-s + (−0.0244 − 0.0753i)21-s + (−1.55 + 4.79i)23-s + (0.309 + 0.951i)25-s − 1.16i·27-s + (1.35 − 1.86i)29-s + (8.10 − 5.88i)31-s + ⋯
L(s)  = 1  − 0.113i·3-s + (−0.361 − 0.262i)5-s + (0.145 − 0.0472i)7-s + 0.987·9-s + (0.391 + 0.539i)11-s + (−0.00854 − 0.00277i)13-s + (−0.0297 + 0.0409i)15-s + (−0.563 − 0.774i)17-s + (0.798 − 0.259i)19-s + (−0.00534 − 0.0164i)21-s + (−0.325 + 1.00i)23-s + (0.0618 + 0.190i)25-s − 0.224i·27-s + (0.251 − 0.345i)29-s + (1.45 − 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.919 + 0.392i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.919 + 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59310 - 0.325336i\)
\(L(\frac12)\) \(\approx\) \(1.59310 - 0.325336i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 + 6.25i)T \)
good3 \( 1 + 0.195iT - 3T^{2} \)
7 \( 1 + (-0.384 + 0.124i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.29 - 1.78i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.0308 + 0.0100i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.32 + 3.19i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.48 + 1.13i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.55 - 4.79i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.35 + 1.86i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-8.10 + 5.88i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-5.75 - 4.18i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-3.35 + 10.3i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-9.03 - 2.93i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.08 + 4.25i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.612 + 1.88i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.03 - 3.18i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.367 - 0.505i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-9.36 - 12.8i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 - 1.73iT - 79T^{2} \)
83 \( 1 + 15.9T + 83T^{2} \)
89 \( 1 + (1.34 - 0.435i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.27 + 3.12i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.921247838317832160952360235096, −9.536343292985920087379169950696, −8.444372444737465857512547525593, −7.44818756404851058134274644319, −6.99233521401297420130676159183, −5.75161867898541029851427285877, −4.61472988391575208064770929148, −3.97803438349654133789783184706, −2.46848523663951999424956776984, −1.04421568834390843593386443736, 1.23198049074417084263496283069, 2.80019714679669599947646195519, 3.99249389307760175624453326064, 4.70820033816642570974993378009, 6.07416557829867540509380263888, 6.78099152270636768808389639909, 7.78767340459447763044832641089, 8.523989975968380784540869289015, 9.507079628085965117264958795063, 10.35552863524141390582453009196

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