Properties

Label 2-820-820.703-c0-0-0
Degree $2$
Conductor $820$
Sign $0.213 + 0.976i$
Analytic cond. $0.409233$
Root an. cond. $0.639713$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.987 − 0.156i)5-s + (0.809 + 0.587i)8-s + (−0.707 − 0.707i)9-s + (−0.453 − 0.891i)10-s + (1.04 + 0.0819i)13-s + (0.309 − 0.951i)16-s + (−0.0366 + 0.152i)17-s + (−0.453 + 0.891i)18-s + (−0.707 + 0.707i)20-s + (0.951 − 0.309i)25-s + (−0.243 − 1.01i)26-s + (0.965 − 1.57i)29-s − 32-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.987 − 0.156i)5-s + (0.809 + 0.587i)8-s + (−0.707 − 0.707i)9-s + (−0.453 − 0.891i)10-s + (1.04 + 0.0819i)13-s + (0.309 − 0.951i)16-s + (−0.0366 + 0.152i)17-s + (−0.453 + 0.891i)18-s + (−0.707 + 0.707i)20-s + (0.951 − 0.309i)25-s + (−0.243 − 1.01i)26-s + (0.965 − 1.57i)29-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.213 + 0.976i$
Analytic conductor: \(0.409233\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :0),\ 0.213 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9088179517\)
\(L(\frac12)\) \(\approx\) \(0.9088179517\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.987 + 0.156i)T \)
41 \( 1 + (0.951 - 0.309i)T \)
good3 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.156 + 0.987i)T^{2} \)
11 \( 1 + (-0.891 - 0.453i)T^{2} \)
13 \( 1 + (-1.04 - 0.0819i)T + (0.987 + 0.156i)T^{2} \)
17 \( 1 + (0.0366 - 0.152i)T + (-0.891 - 0.453i)T^{2} \)
19 \( 1 + (0.156 + 0.987i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (-0.965 + 1.57i)T + (-0.453 - 0.891i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.183 - 1.16i)T + (-0.951 - 0.309i)T^{2} \)
43 \( 1 + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (-0.156 + 0.987i)T^{2} \)
53 \( 1 + (1.47 - 0.355i)T + (0.891 - 0.453i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \)
67 \( 1 + (-0.453 - 0.891i)T^{2} \)
71 \( 1 + (0.891 + 0.453i)T^{2} \)
73 \( 1 - 1.97iT - T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.0600 + 0.763i)T + (-0.987 + 0.156i)T^{2} \)
97 \( 1 + (1.01 - 1.65i)T + (-0.453 - 0.891i)T^{2} \)
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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

−10.12887916109708358454323662838, −9.601121121429279537621486445146, −8.665119732575540643950911013321, −8.219142293589197658586500312704, −6.63193702142020376304542314024, −5.85870483155783788970824790394, −4.74679489281976994680127300412, −3.56379226551010250209223629259, −2.57124118521226436510873457139, −1.27181600472991252701223848036, 1.63097389078000598161265463394, 3.19283982782157259700036785662, 4.74445791954053767548113811153, 5.53916357185026516771396152034, 6.25112226449356288804314478573, 7.08860552734503423926378665538, 8.184014396836088826880635672376, 8.822549529279414544302442122969, 9.560592906656832455784713904717, 10.64234902796131964958476024534

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