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Properties

Label 2-820-41.21-c1-0-7
Degree $2$
Conductor $820$
Sign $0.999 - 0.0226i$
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  + (0.901 + 0.901i)3-s + (0.587 − 0.809i)5-s + (−4.93e−6 + 9.69e−6i)7-s − 1.37i·9-s + (2.30 + 0.364i)11-s + (3.08 − 1.57i)13-s + (1.25 − 0.199i)15-s + (−0.986 + 6.23i)17-s + (−1.02 − 0.521i)19-s + (−1.31e−5 + 4.28e−6i)21-s + (2.63 − 8.10i)23-s + (−0.309 − 0.951i)25-s + (3.94 − 3.94i)27-s + (0.446 + 2.82i)29-s + (2.29 − 1.66i)31-s + ⋯
L(s)  = 1  + (0.520 + 0.520i)3-s + (0.262 − 0.361i)5-s + (−1.86e−6 + 3.66e−6i)7-s − 0.458i·9-s + (0.693 + 0.109i)11-s + (0.855 − 0.436i)13-s + (0.325 − 0.0515i)15-s + (−0.239 + 1.51i)17-s + (−0.234 − 0.119i)19-s − 2.87e − 6·21-s + (0.548 − 1.68i)23-s + (−0.0618 − 0.190i)25-s + (0.758 − 0.758i)27-s + (0.0829 + 0.523i)29-s + (0.412 − 0.299i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.999 - 0.0226i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.999 - 0.0226i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06644 + 0.0234129i\)
\(L(\frac12)\) \(\approx\) \(2.06644 + 0.0234129i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (3.81 + 5.14i)T \)
good3 \( 1 + (-0.901 - 0.901i)T + 3iT^{2} \)
7 \( 1 + (4.93e-6 - 9.69e-6i)T + (-4.11 - 5.66i)T^{2} \)
11 \( 1 + (-2.30 - 0.364i)T + (10.4 + 3.39i)T^{2} \)
13 \( 1 + (-3.08 + 1.57i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.986 - 6.23i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (1.02 + 0.521i)T + (11.1 + 15.3i)T^{2} \)
23 \( 1 + (-2.63 + 8.10i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.446 - 2.82i)T + (-27.5 + 8.96i)T^{2} \)
31 \( 1 + (-2.29 + 1.66i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-6.84 - 4.97i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (0.267 + 0.0870i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (-1.68 - 3.30i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-1.06 - 6.72i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (0.325 - 1.00i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.43 - 2.41i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (10.6 - 1.68i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (0.388 + 0.0615i)T + (67.5 + 21.9i)T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 + (3.38 + 3.38i)T + 79iT^{2} \)
83 \( 1 - 5.94T + 83T^{2} \)
89 \( 1 + (7.18 - 14.0i)T + (-52.3 - 72.0i)T^{2} \)
97 \( 1 + (-2.31 + 0.366i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.33297308613111661578346545889, −9.104060463114557067374488481155, −8.819672172606339826099473698468, −7.960262710287356029692703903350, −6.50527730849519825611236655767, −6.05621563666436508609508980840, −4.57397639563066571170452860409, −3.91332072123984375134409445050, −2.78345458736561269118047188517, −1.21803359428788332954828901360, 1.39851408125496142894514529573, 2.56137669513947499261311801709, 3.61708777866798886091321181784, 4.87653281671326883889782480270, 5.97743921311921735354739022544, 6.94217841707013423862551252525, 7.54879100640209237215901933929, 8.592236693956554388892959976922, 9.275342902731181921058207603952, 10.11949841047966612949051022026

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