Properties

Label 2-80-5.2-c4-0-2
Degree $2$
Conductor $80$
Sign $-0.489 - 0.871i$
Analytic cond. $8.26959$
Root an. cond. $2.87569$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.958 − 0.958i)3-s + (−1.04 + 24.9i)5-s + (−26.0 − 26.0i)7-s + 79.1i·9-s − 75.9·11-s + (−99.1 + 99.1i)13-s + (22.9 + 24.9i)15-s + (77.9 + 77.9i)17-s + 640. i·19-s − 49.9·21-s + (−117. + 117. i)23-s + (−622. − 52.0i)25-s + (153. + 153. i)27-s − 752. i·29-s − 708.·31-s + ⋯
L(s)  = 1  + (0.106 − 0.106i)3-s + (−0.0416 + 0.999i)5-s + (−0.531 − 0.531i)7-s + 0.977i·9-s − 0.627·11-s + (−0.586 + 0.586i)13-s + (0.101 + 0.110i)15-s + (0.269 + 0.269i)17-s + 1.77i·19-s − 0.113·21-s + (−0.221 + 0.221i)23-s + (−0.996 − 0.0832i)25-s + (0.210 + 0.210i)27-s − 0.894i·29-s − 0.736·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.871i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.489 - 0.871i$
Analytic conductor: \(8.26959\)
Root analytic conductor: \(2.87569\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :2),\ -0.489 - 0.871i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.527708 + 0.901789i\)
\(L(\frac12)\) \(\approx\) \(0.527708 + 0.901789i\)
\(L(3)\) 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 \( 1 + (1.04 - 24.9i)T \)
good3 \( 1 + (-0.958 + 0.958i)T - 81iT^{2} \)
7 \( 1 + (26.0 + 26.0i)T + 2.40e3iT^{2} \)
11 \( 1 + 75.9T + 1.46e4T^{2} \)
13 \( 1 + (99.1 - 99.1i)T - 2.85e4iT^{2} \)
17 \( 1 + (-77.9 - 77.9i)T + 8.35e4iT^{2} \)
19 \( 1 - 640. iT - 1.30e5T^{2} \)
23 \( 1 + (117. - 117. i)T - 2.79e5iT^{2} \)
29 \( 1 + 752. iT - 7.07e5T^{2} \)
31 \( 1 + 708.T + 9.23e5T^{2} \)
37 \( 1 + (-1.17e3 - 1.17e3i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.82e3T + 2.82e6T^{2} \)
43 \( 1 + (-2.37e3 + 2.37e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (2.45e3 + 2.45e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-2.11e3 + 2.11e3i)T - 7.89e6iT^{2} \)
59 \( 1 - 5.25e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.31e3T + 1.38e7T^{2} \)
67 \( 1 + (-5.56e3 - 5.56e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 7.55e3T + 2.54e7T^{2} \)
73 \( 1 + (-1.20e3 + 1.20e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 3.39e3iT - 3.89e7T^{2} \)
83 \( 1 + (928. - 928. i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.19e4iT - 6.27e7T^{2} \)
97 \( 1 + (-4.68e3 - 4.68e3i)T + 8.85e7iT^{2} \)
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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

−14.02220187409144363120295706550, −13.10308801944172424054691363105, −11.76664618023962405459371646664, −10.50999703343274851843753675985, −9.913741254974132819580436701332, −8.008755714615810080742866102064, −7.16989957992045042142017976932, −5.75572365685735759473980997616, −3.88396801592360327910131518632, −2.28290608373619159146096367311, 0.49920449577049735180187990288, 2.87798910150210049080076702484, 4.69039238448878153584298311987, 5.95810166984472806623665259098, 7.56490303886254670391438860643, 9.006660288490275853507381135416, 9.574828850261999111891192215249, 11.18722451514396397444311028041, 12.59481689992552011854242439442, 12.84887932471180872111723350584

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