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Properties

Label 2-820-41.23-c1-0-5
Degree $2$
Conductor $820$
Sign $0.606 - 0.794i$
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  + 2.07i·3-s + (0.809 − 0.587i)5-s + (0.0731 + 0.0237i)7-s − 1.32·9-s + (1.84 − 2.53i)11-s + (3.95 − 1.28i)13-s + (1.22 + 1.68i)15-s + (−0.197 + 0.271i)17-s + (4.57 + 1.48i)19-s + (−0.0494 + 0.152i)21-s + (0.630 + 1.93i)23-s + (0.309 − 0.951i)25-s + 3.48i·27-s + (0.397 + 0.547i)29-s + (0.131 + 0.0952i)31-s + ⋯
L(s)  = 1  + 1.20i·3-s + (0.361 − 0.262i)5-s + (0.0276 + 0.00898i)7-s − 0.440·9-s + (0.555 − 0.764i)11-s + (1.09 − 0.356i)13-s + (0.315 + 0.434i)15-s + (−0.0479 + 0.0659i)17-s + (1.04 + 0.341i)19-s + (−0.0107 + 0.0331i)21-s + (0.131 + 0.404i)23-s + (0.0618 − 0.190i)25-s + 0.671i·27-s + (0.0738 + 0.101i)29-s + (0.0235 + 0.0171i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.606 - 0.794i$
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: \((2,\ 820,\ (\ :1/2),\ 0.606 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65011 + 0.816305i\)
\(L(\frac12)\) \(\approx\) \(1.65011 + 0.816305i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.29 + 4.75i)T \)
good3 \( 1 - 2.07iT - 3T^{2} \)
7 \( 1 + (-0.0731 - 0.0237i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.84 + 2.53i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.95 + 1.28i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.197 - 0.271i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.57 - 1.48i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.630 - 1.93i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.397 - 0.547i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.131 - 0.0952i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.30 - 2.39i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-1.63 - 5.01i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (3.32 - 1.07i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.511 + 0.704i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.88 - 5.79i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.43 + 7.48i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-4.20 - 5.78i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (1.33 - 1.83i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 - 4.33T + 73T^{2} \)
79 \( 1 - 5.37iT - 79T^{2} \)
83 \( 1 - 3.17T + 83T^{2} \)
89 \( 1 + (-1.00 - 0.327i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.76 - 3.81i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.25529360910197841713292347288, −9.567418522887018335825250929799, −8.845053941881673103121455524278, −8.091274416740656873128577271869, −6.74937898459031123335556606103, −5.73708677302308270237327021646, −5.04728366458291345032405776050, −3.86625699306680506610518104796, −3.23794150688236200002109641225, −1.32553996393154636595918798371, 1.18991326358689518730581491407, 2.14260548756669466010102869996, 3.50057702432581997737591590788, 4.79320660709282631369252351021, 6.03818428081007362665915209824, 6.72883883407564143562920021768, 7.34161771555900634093121600413, 8.303593068766552484964000603065, 9.227347953879929932170652615594, 10.04905744125649659821335099274

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