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Properties

Label 2-820-41.23-c1-0-11
Degree $2$
Conductor $820$
Sign $-0.0758 + 0.997i$
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  − 0.525i·3-s + (0.809 − 0.587i)5-s + (−3.06 − 0.995i)7-s + 2.72·9-s + (0.883 − 1.21i)11-s + (−1.58 + 0.515i)13-s + (−0.309 − 0.425i)15-s + (1.65 − 2.27i)17-s + (2.25 + 0.732i)19-s + (−0.523 + 1.61i)21-s + (−2.75 − 8.48i)23-s + (0.309 − 0.951i)25-s − 3.00i·27-s + (−0.808 − 1.11i)29-s + (−4.19 − 3.04i)31-s + ⋯
L(s)  = 1  − 0.303i·3-s + (0.361 − 0.262i)5-s + (−1.15 − 0.376i)7-s + 0.907·9-s + (0.266 − 0.366i)11-s + (−0.440 + 0.143i)13-s + (−0.0798 − 0.109i)15-s + (0.401 − 0.552i)17-s + (0.517 + 0.168i)19-s + (−0.114 + 0.351i)21-s + (−0.575 − 1.77i)23-s + (0.0618 − 0.190i)25-s − 0.579i·27-s + (−0.150 − 0.206i)29-s + (−0.753 − 0.547i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0758 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.0758 + 0.997i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.0758 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.919461 - 0.992084i\)
\(L(\frac12)\) \(\approx\) \(0.919461 - 0.992084i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.18 - 1.64i)T \)
good3 \( 1 + 0.525iT - 3T^{2} \)
7 \( 1 + (3.06 + 0.995i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.883 + 1.21i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.58 - 0.515i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.65 + 2.27i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.25 - 0.732i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.75 + 8.48i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.808 + 1.11i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.19 + 3.04i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.95 - 1.42i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (2.00 + 6.15i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.01 + 0.654i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.05 + 2.83i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.735 + 2.26i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.906 + 2.78i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-6.52 - 8.98i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (0.738 - 1.01i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 - 4.14T + 73T^{2} \)
79 \( 1 - 0.197iT - 79T^{2} \)
83 \( 1 - 0.156T + 83T^{2} \)
89 \( 1 + (9.00 + 2.92i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.51 - 4.83i)T + (-29.9 + 92.2i)T^{2} \)
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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

−9.864910252946856769663650072996, −9.428456689958536991651464677784, −8.299393922215925191794445082674, −7.24892246395096081041703488188, −6.64095302339131745521104735455, −5.74421498228442559888836366679, −4.52714541391926467712524000628, −3.53688408943788931275120997366, −2.24983948814070010553431282804, −0.68262849538318744889616545029, 1.66860745267691280102581434863, 3.10297579569898446339592461493, 3.93887251295929518776057675104, 5.23176614827712557306988242004, 6.07235269734695613022105167305, 7.02875392625509984295088082859, 7.70616022507005522247495974506, 9.186479300300532902516141562144, 9.615491933214928655573560724485, 10.19707509752412465312751823961

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