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Properties

Label 2-820-820.567-c0-0-0
Degree $2$
Conductor $820$
Sign $0.569 - 0.821i$
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 + (0.133 + 1.70i)13-s + (0.309 + 0.951i)16-s + (1.93 − 0.465i)17-s + (0.453 + 0.891i)18-s + (0.707 + 0.707i)20-s + (0.951 + 0.309i)25-s + (−1.65 − 0.398i)26-s + (0.652 − 0.399i)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 + (0.133 + 1.70i)13-s + (0.309 + 0.951i)16-s + (1.93 − 0.465i)17-s + (0.453 + 0.891i)18-s + (0.707 + 0.707i)20-s + (0.951 + 0.309i)25-s + (−1.65 − 0.398i)26-s + (0.652 − 0.399i)29-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.569 - 0.821i$
Analytic conductor: \(0.409233\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :0),\ 0.569 - 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7287761932\)
\(L(\frac12)\) \(\approx\) \(0.7287761932\)
\(L(1)\) 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 \( 1 + (-0.133 - 1.70i)T + (-0.987 + 0.156i)T^{2} \)
17 \( 1 + (-1.93 + 0.465i)T + (0.891 - 0.453i)T^{2} \)
19 \( 1 + (-0.156 + 0.987i)T^{2} \)
23 \( 1 + (-0.587 + 0.809i)T^{2} \)
29 \( 1 + (-0.652 + 0.399i)T + (0.453 - 0.891i)T^{2} \)
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 \( 1 + (-0.303 + 1.26i)T + (-0.891 - 0.453i)T^{2} \)
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 \( 1 + (1.84 + 0.144i)T + (0.987 + 0.156i)T^{2} \)
97 \( 1 + (-0.398 + 0.243i)T + (0.453 - 0.891i)T^{2} \)
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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

−10.18428623017912794046473147248, −9.634115549063491364589336387765, −8.721986718177752551528806910126, −7.973220501093206236218622915126, −7.06022994893225173447651406436, −6.58416489369379159996965082890, −5.27527190839665025672644279454, −4.33574359730791331032101558094, −3.55229591874200797893059816810, −1.23014123901706028452141034605, 1.17273397635907200023516797861, 2.88799173453384938105111012110, 3.62970773009856894796966318490, 4.69970221044381595485913760368, 5.65978116925194618029201422073, 7.41564742261905682728556363690, 7.82548161625353222668201045723, 8.512096362301687649708298055195, 9.762277143040011388928567831006, 10.54151890923225680614430971248

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