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Properties

Label 2-820-205.189-c1-0-8
Degree $2$
Conductor $820$
Sign $0.999 + 0.0188i$
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  − 3.03·3-s + (1.66 + 1.49i)5-s + (−0.0669 − 0.205i)7-s + 6.20·9-s + (−2.13 − 2.93i)11-s + (0.699 − 2.15i)13-s + (−5.04 − 4.53i)15-s + (−2.80 + 2.04i)17-s + (−2.36 + 0.769i)19-s + (0.202 + 0.624i)21-s + (5.90 + 1.91i)23-s + (0.529 + 4.97i)25-s − 9.71·27-s + (3.38 − 4.66i)29-s + (−0.260 + 0.189i)31-s + ⋯
L(s)  = 1  − 1.75·3-s + (0.743 + 0.668i)5-s + (−0.0252 − 0.0778i)7-s + 2.06·9-s + (−0.642 − 0.884i)11-s + (0.194 − 0.597i)13-s + (−1.30 − 1.17i)15-s + (−0.681 + 0.495i)17-s + (−0.543 + 0.176i)19-s + (0.0442 + 0.136i)21-s + (1.23 + 0.400i)23-s + (0.105 + 0.994i)25-s − 1.86·27-s + (0.629 − 0.866i)29-s + (−0.0467 + 0.0339i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.999 + 0.0188i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.999 + 0.0188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904293 - 0.00854496i\)
\(L(\frac12)\) \(\approx\) \(0.904293 - 0.00854496i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 - 1.49i)T \)
41 \( 1 + (-1.70 - 6.17i)T \)
good3 \( 1 + 3.03T + 3T^{2} \)
7 \( 1 + (0.0669 + 0.205i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (2.13 + 2.93i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.699 + 2.15i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.80 - 2.04i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.36 - 0.769i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-5.90 - 1.91i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-3.38 + 4.66i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.260 - 0.189i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-4.68 + 6.44i)T + (-11.4 - 35.1i)T^{2} \)
43 \( 1 + (-5.69 - 1.84i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (-3.16 + 9.73i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-8.51 - 6.18i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.33 + 4.09i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.56 + 4.80i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-5.42 - 3.94i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-9.55 - 13.1i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 - 6.64iT - 73T^{2} \)
79 \( 1 - 0.296iT - 79T^{2} \)
83 \( 1 - 3.09iT - 83T^{2} \)
89 \( 1 + (9.05 - 2.94i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.18 + 2.31i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.51448217838790896207809459401, −9.781341875348911271783464797272, −8.548628998616177906403065143533, −7.32259125830163004824742658976, −6.50363298006209596501115180558, −5.80914867170361687007748894921, −5.28647910760383648913175421039, −4.01338635211493885369386207482, −2.51083012183832093748996189932, −0.807681549200446556611073804517, 0.904445922438422282784008151643, 2.29399595064939332739324181572, 4.51553801668741619392706355283, 4.85322542911523016600859908232, 5.79975774408870139405169925611, 6.59679662958582674347457515015, 7.29933494428503577458769284395, 8.779769643902544767146700188444, 9.521085860205693860357053946409, 10.48446561460876696774201504248

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