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Properties

Label 2-80-80.29-c5-0-11
Degree $2$
Conductor $80$
Sign $-0.421 - 0.907i$
Analytic cond. $12.8307$
Root an. cond. $3.58199$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.44 + 1.54i)2-s + (9.88 − 9.88i)3-s + (27.2 − 16.8i)4-s + (27.0 + 48.9i)5-s + (−38.4 + 69.0i)6-s − 163.·7-s + (−122. + 133. i)8-s + 47.6i·9-s + (−222. − 224. i)10-s + (237. − 237. i)11-s + (102. − 435. i)12-s + (−496. + 496. i)13-s + (891. − 253. i)14-s + (750. + 216. i)15-s + (457. − 916. i)16-s + 595. i·17-s + ⋯
L(s)  = 1  + (−0.961 + 0.273i)2-s + (0.633 − 0.633i)3-s + (0.850 − 0.526i)4-s + (0.483 + 0.875i)5-s + (−0.436 + 0.783i)6-s − 1.26·7-s + (−0.674 + 0.738i)8-s + 0.196i·9-s + (−0.704 − 0.709i)10-s + (0.591 − 0.591i)11-s + (0.205 − 0.872i)12-s + (−0.814 + 0.814i)13-s + (1.21 − 0.345i)14-s + (0.861 + 0.248i)15-s + (0.446 − 0.894i)16-s + 0.499i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.421 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.421 - 0.907i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.421 - 0.907i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.457830 + 0.717340i\)
\(L(\frac12)\) \(\approx\) \(0.457830 + 0.717340i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.44 - 1.54i)T \)
5 \( 1 + (-27.0 - 48.9i)T \)
good3 \( 1 + (-9.88 + 9.88i)T - 243iT^{2} \)
7 \( 1 + 163.T + 1.68e4T^{2} \)
11 \( 1 + (-237. + 237. i)T - 1.61e5iT^{2} \)
13 \( 1 + (496. - 496. i)T - 3.71e5iT^{2} \)
17 \( 1 - 595. iT - 1.41e6T^{2} \)
19 \( 1 + (-424. - 424. i)T + 2.47e6iT^{2} \)
23 \( 1 + 4.15e3T + 6.43e6T^{2} \)
29 \( 1 + (-3.20e3 - 3.20e3i)T + 2.05e7iT^{2} \)
31 \( 1 + 7.96e3T + 2.86e7T^{2} \)
37 \( 1 + (-7.88e3 - 7.88e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.08e4iT - 1.15e8T^{2} \)
43 \( 1 + (-1.48e4 - 1.48e4i)T + 1.47e8iT^{2} \)
47 \( 1 + 1.02e4iT - 2.29e8T^{2} \)
53 \( 1 + (2.54e4 + 2.54e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-1.43e3 + 1.43e3i)T - 7.14e8iT^{2} \)
61 \( 1 + (6.07e3 + 6.07e3i)T + 8.44e8iT^{2} \)
67 \( 1 + (-2.79e4 + 2.79e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 1.71e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.24e4T + 2.07e9T^{2} \)
79 \( 1 - 3.18e3T + 3.07e9T^{2} \)
83 \( 1 + (2.45e4 - 2.45e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 3.64e4iT - 5.58e9T^{2} \)
97 \( 1 + 3.26e4iT - 8.58e9T^{2} \)
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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

−14.07501308009886086879693444159, −12.73819483086519973751897200144, −11.35622115932611868951542057978, −10.07133251560318682870188085019, −9.340387048201476056107221756758, −7.979738643725984897666673913005, −6.85913057913821309452458418346, −6.11501723829302829173261870048, −3.09274302985514218262209959258, −1.83773783046583910379472406503, 0.42774196482001562763766044450, 2.48500385534271417897190688449, 3.95088542002814073708846642250, 6.04872819369030391884612499085, 7.55002207846201903728840729277, 9.047136093963078181010565406313, 9.537033776402732960867688038040, 10.23069984704868586490081395878, 12.14011010667305793897518679979, 12.70844636182586706471254822411

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