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Properties

Label 2-820-41.4-c1-0-7
Degree $2$
Conductor $820$
Sign $0.755 - 0.655i$
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  + 1.67i·3-s + (−0.309 + 0.951i)5-s + (1.17 − 1.62i)7-s + 0.195·9-s + (5.72 − 1.85i)11-s + (−1.32 − 1.82i)13-s + (−1.59 − 0.517i)15-s + (3.45 − 1.12i)17-s + (0.272 − 0.374i)19-s + (2.71 + 1.97i)21-s + (0.446 − 0.324i)23-s + (−0.809 − 0.587i)25-s + 5.35i·27-s + (−2.29 − 0.746i)29-s + (0.721 + 2.22i)31-s + ⋯
L(s)  = 1  + 0.966i·3-s + (−0.138 + 0.425i)5-s + (0.445 − 0.613i)7-s + 0.0652·9-s + (1.72 − 0.560i)11-s + (−0.367 − 0.506i)13-s + (−0.411 − 0.133i)15-s + (0.838 − 0.272i)17-s + (0.0624 − 0.0859i)19-s + (0.593 + 0.431i)21-s + (0.0930 − 0.0675i)23-s + (−0.161 − 0.117i)25-s + 1.02i·27-s + (−0.426 − 0.138i)29-s + (0.129 + 0.399i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69982 + 0.634798i\)
\(L(\frac12)\) \(\approx\) \(1.69982 + 0.634798i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.69 - 2.92i)T \)
good3 \( 1 - 1.67iT - 3T^{2} \)
7 \( 1 + (-1.17 + 1.62i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-5.72 + 1.85i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.32 + 1.82i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.45 + 1.12i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.272 + 0.374i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.446 + 0.324i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.29 + 0.746i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.721 - 2.22i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.487 + 1.50i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (5.16 - 3.75i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (1.55 + 2.14i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.24 + 1.70i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.95 - 1.42i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.95 - 7.23i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (9.54 + 3.10i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (9.70 - 3.15i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 + 2.67iT - 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + (-2.25 + 3.10i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.11 - 1.01i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.24788367702321138654000641126, −9.638836094244891890041108526453, −8.822435096666277897373309000321, −7.72680288662446861211558292446, −6.93847330556554754339790064138, −5.89195485776264819703931887217, −4.74702909543713699449120987244, −3.94834278607379831900060887141, −3.16552941630428175973025812903, −1.24752722564846521303677261653, 1.26204822235554430455985154052, 2.06496462268166051169596189240, 3.76954134355441086771330607533, 4.73781904423502522692895489639, 5.90549476533501558331137657058, 6.76948718947408520508661723545, 7.50223874756280377914930038511, 8.380422220117845268596766377337, 9.245922710384347059773909090829, 9.905614006854783769748220299555

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