Properties

Label 2-80-80.27-c1-0-4
Degree $2$
Conductor $80$
Sign $0.596 - 0.802i$
Analytic cond. $0.638803$
Root an. cond. $0.799251$
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  + (0.567 + 1.29i)2-s + 1.96·3-s + (−1.35 + 1.47i)4-s + (−1.42 − 1.72i)5-s + (1.11 + 2.54i)6-s + (−1.60 − 1.60i)7-s + (−2.67 − 0.920i)8-s + 0.851·9-s + (1.42 − 2.82i)10-s + (0.754 − 0.754i)11-s + (−2.65 + 2.88i)12-s + 5.94i·13-s + (1.16 − 2.98i)14-s + (−2.79 − 3.38i)15-s + (−0.327 − 3.98i)16-s + (1.95 + 1.95i)17-s + ⋯
L(s)  = 1  + (0.401 + 0.915i)2-s + 1.13·3-s + (−0.677 + 0.735i)4-s + (−0.635 − 0.771i)5-s + (0.454 + 1.03i)6-s + (−0.605 − 0.605i)7-s + (−0.945 − 0.325i)8-s + 0.283·9-s + (0.451 − 0.892i)10-s + (0.227 − 0.227i)11-s + (−0.767 + 0.833i)12-s + 1.64i·13-s + (0.311 − 0.797i)14-s + (−0.720 − 0.874i)15-s + (−0.0817 − 0.996i)16-s + (0.474 + 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.596 - 0.802i$
Analytic conductor: \(0.638803\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1/2),\ 0.596 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12326 + 0.564837i\)
\(L(\frac12)\) \(\approx\) \(1.12326 + 0.564837i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.567 - 1.29i)T \)
5 \( 1 + (1.42 + 1.72i)T \)
good3 \( 1 - 1.96T + 3T^{2} \)
7 \( 1 + (1.60 + 1.60i)T + 7iT^{2} \)
11 \( 1 + (-0.754 + 0.754i)T - 11iT^{2} \)
13 \( 1 - 5.94iT - 13T^{2} \)
17 \( 1 + (-1.95 - 1.95i)T + 17iT^{2} \)
19 \( 1 + (-0.780 + 0.780i)T - 19iT^{2} \)
23 \( 1 + (-4.93 + 4.93i)T - 23iT^{2} \)
29 \( 1 + (1.44 + 1.44i)T + 29iT^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 - 10.2iT - 37T^{2} \)
41 \( 1 + 6.93iT - 41T^{2} \)
43 \( 1 - 9.91iT - 43T^{2} \)
47 \( 1 + (-0.104 + 0.104i)T - 47iT^{2} \)
53 \( 1 + 4.03T + 53T^{2} \)
59 \( 1 + (3.46 + 3.46i)T + 59iT^{2} \)
61 \( 1 + (-0.680 + 0.680i)T - 61iT^{2} \)
67 \( 1 + 9.04iT - 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 + (2.94 + 2.94i)T + 73iT^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 4.23T + 83T^{2} \)
89 \( 1 - 0.0426T + 89T^{2} \)
97 \( 1 + (1.91 + 1.91i)T + 97iT^{2} \)
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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

−14.52206035123816024138846833875, −13.68410019521263441384548967045, −12.86144039597457212956680145597, −11.63120082817382229975264835034, −9.478348463306719747871941668958, −8.738446697706000914364511974823, −7.73806664801300778972291765843, −6.53130278658372153032073880351, −4.55247916818061264944834478904, −3.46424998881996386976603221864, 2.81800956654214632124595173369, 3.49156147385990727870900608353, 5.59605059758120906826903241569, 7.50873187294690432622234878535, 8.813029513449388483042429036768, 9.839318463301719580249645007908, 10.98657651729292188774579293443, 12.19510876943067033163888416241, 13.12694252970750213745510856233, 14.24802322010725185251450079356

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