Properties

Label 2-80-5.2-c8-0-12
Degree $2$
Conductor $80$
Sign $0.506 + 0.862i$
Analytic cond. $32.5902$
Root an. cond. $5.70879$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−75.2 + 75.2i)3-s + (−14.1 + 624. i)5-s + (−730. − 730. i)7-s − 4.77e3i·9-s − 1.95e4·11-s + (−2.49e4 + 2.49e4i)13-s + (−4.59e4 − 4.81e4i)15-s + (−1.12e4 − 1.12e4i)17-s + 1.71e5i·19-s + 1.10e5·21-s + (1.32e5 − 1.32e5i)23-s + (−3.90e5 − 1.77e4i)25-s + (−1.34e5 − 1.34e5i)27-s − 1.27e5i·29-s + 9.60e5·31-s + ⋯
L(s)  = 1  + (−0.929 + 0.929i)3-s + (−0.0226 + 0.999i)5-s + (−0.304 − 0.304i)7-s − 0.728i·9-s − 1.33·11-s + (−0.872 + 0.872i)13-s + (−0.908 − 0.950i)15-s + (−0.135 − 0.135i)17-s + 1.31i·19-s + 0.566·21-s + (0.471 − 0.471i)23-s + (−0.998 − 0.0453i)25-s + (−0.252 − 0.252i)27-s − 0.179i·29-s + 1.04·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(32.5902\)
Root analytic conductor: \(5.70879\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :4),\ 0.506 + 0.862i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0505181 - 0.0289212i\)
\(L(\frac12)\) \(\approx\) \(0.0505181 - 0.0289212i\)
\(L(5)\) 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 \( 1 + (14.1 - 624. i)T \)
good3 \( 1 + (75.2 - 75.2i)T - 6.56e3iT^{2} \)
7 \( 1 + (730. + 730. i)T + 5.76e6iT^{2} \)
11 \( 1 + 1.95e4T + 2.14e8T^{2} \)
13 \( 1 + (2.49e4 - 2.49e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (1.12e4 + 1.12e4i)T + 6.97e9iT^{2} \)
19 \( 1 - 1.71e5iT - 1.69e10T^{2} \)
23 \( 1 + (-1.32e5 + 1.32e5i)T - 7.83e10iT^{2} \)
29 \( 1 + 1.27e5iT - 5.00e11T^{2} \)
31 \( 1 - 9.60e5T + 8.52e11T^{2} \)
37 \( 1 + (2.43e5 + 2.43e5i)T + 3.51e12iT^{2} \)
41 \( 1 - 2.50e6T + 7.98e12T^{2} \)
43 \( 1 + (-6.76e3 + 6.76e3i)T - 1.16e13iT^{2} \)
47 \( 1 + (-1.79e6 - 1.79e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (2.97e6 - 2.97e6i)T - 6.22e13iT^{2} \)
59 \( 1 + 3.13e5iT - 1.46e14T^{2} \)
61 \( 1 - 1.76e7T + 1.91e14T^{2} \)
67 \( 1 + (4.41e6 + 4.41e6i)T + 4.06e14iT^{2} \)
71 \( 1 - 8.89e6T + 6.45e14T^{2} \)
73 \( 1 + (-1.95e7 + 1.95e7i)T - 8.06e14iT^{2} \)
79 \( 1 - 1.11e7iT - 1.51e15T^{2} \)
83 \( 1 + (1.58e7 - 1.58e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 4.85e7iT - 3.93e15T^{2} \)
97 \( 1 + (1.07e8 + 1.07e8i)T + 7.83e15iT^{2} \)
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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

−12.30508358540888870546679333240, −11.18845661260771095508861246027, −10.37552123030360057212856919312, −9.750655759135497441925957573801, −7.79583394226201046953159953679, −6.54553505705720774132572717519, −5.31963634276963115047610183701, −4.09517441880961562353551136064, −2.53805152763485066221945326126, −0.02626370864021809026734859977, 0.867108337909376549697093389960, 2.57958159985600960044881916329, 4.92613627391554804561551871827, 5.68699365601840398591832796249, 7.11244887038936403794091307687, 8.179971708402015616785197619103, 9.584387931677775788583162812422, 10.94834952883202483137736285136, 12.10232686862502035264521754992, 12.84126671299566647855113493687

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