Properties

Degree $2$
Conductor $80$
Sign $-0.692 + 0.721i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.62 + 0.583i)2-s + (−13.2 + 13.2i)3-s + (31.3 − 6.56i)4-s + (−17.6 − 17.6i)5-s + (66.5 − 81.9i)6-s + 190. i·7-s + (−172. + 55.2i)8-s − 105. i·9-s + (109. + 89.1i)10-s + (436. + 436. i)11-s + (−326. + 500. i)12-s + (−460. + 460. i)13-s + (−110. − 1.06e3i)14-s + 466.·15-s + (937. − 411. i)16-s − 541.·17-s + ⋯
L(s)  = 1  + (−0.994 + 0.103i)2-s + (−0.846 + 0.846i)3-s + (0.978 − 0.205i)4-s + (−0.316 − 0.316i)5-s + (0.754 − 0.929i)6-s + 1.46i·7-s + (−0.952 + 0.305i)8-s − 0.434i·9-s + (0.347 + 0.281i)10-s + (1.08 + 1.08i)11-s + (−0.655 + 1.00i)12-s + (−0.755 + 0.755i)13-s + (−0.151 − 1.45i)14-s + 0.535·15-s + (0.915 − 0.401i)16-s − 0.454·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.692 + 0.721i$
Motivic weight: \(5\)
Character: $\chi_{80} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.692 + 0.721i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.120698 - 0.283070i\)
\(L(\frac12)\) \(\approx\) \(0.120698 - 0.283070i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.62 - 0.583i)T \)
5 \( 1 + (17.6 + 17.6i)T \)
good3 \( 1 + (13.2 - 13.2i)T - 243iT^{2} \)
7 \( 1 - 190. iT - 1.68e4T^{2} \)
11 \( 1 + (-436. - 436. i)T + 1.61e5iT^{2} \)
13 \( 1 + (460. - 460. i)T - 3.71e5iT^{2} \)
17 \( 1 + 541.T + 1.41e6T^{2} \)
19 \( 1 + (-317. + 317. i)T - 2.47e6iT^{2} \)
23 \( 1 + 2.55e3iT - 6.43e6T^{2} \)
29 \( 1 + (690. - 690. i)T - 2.05e7iT^{2} \)
31 \( 1 + 6.61e3T + 2.86e7T^{2} \)
37 \( 1 + (-2.98e3 - 2.98e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.47e4iT - 1.15e8T^{2} \)
43 \( 1 + (-783. - 783. i)T + 1.47e8iT^{2} \)
47 \( 1 - 1.79e4T + 2.29e8T^{2} \)
53 \( 1 + (-5.90e3 - 5.90e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (1.46e4 + 1.46e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (3.46e4 - 3.46e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (4.87e4 - 4.87e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 1.89e4iT - 1.80e9T^{2} \)
73 \( 1 + 8.17e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.76e4T + 3.07e9T^{2} \)
83 \( 1 + (-4.02e4 + 4.02e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 2.02e4iT - 5.58e9T^{2} \)
97 \( 1 + 7.68e4T + 8.58e9T^{2} \)
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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

−14.72554910163756200106793883682, −12.20583500488603662653159000325, −11.89785784958607906686071537683, −10.71101054649385904511667335559, −9.429970820246574867928305534719, −8.919086665746934721362052217750, −7.16476013474922243871125678834, −5.83840468238462626876277321454, −4.54552043483232414255382141425, −2.10348975124331922534657666519, 0.22260287238107919254664889663, 1.22088651168743455324110389269, 3.55210241477757866378511388963, 6.01798337639876914712698208889, 7.06898775265608881294224597727, 7.76241715043892187069957291594, 9.419176648277325542775237126077, 10.77836322263317739351033400498, 11.35931115160668597040490269742, 12.39633438578839951640874739368

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