Properties

Degree $2$
Conductor $80$
Sign $-0.192 + 0.981i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−38 + 41i)5-s − 243i·9-s + (475 − 475i)13-s + (−1.52e3 − 1.52e3i)17-s + (−237 − 3.11e3i)25-s − 8.56e3i·29-s + (−475 − 475i)37-s − 4.95e3·41-s + (9.96e3 + 9.23e3i)45-s + 1.68e4i·49-s + (−1.64e4 + 1.64e4i)53-s + 5.49e4·61-s + (1.42e3 + 3.75e4i)65-s + (−5.44e4 + 5.44e4i)73-s − 5.90e4·81-s + ⋯
L(s)  = 1  + (−0.679 + 0.733i)5-s i·9-s + (0.779 − 0.779i)13-s + (−1.27 − 1.27i)17-s + (−0.0758 − 0.997i)25-s − 1.89i·29-s + (−0.0570 − 0.0570i)37-s − 0.460·41-s + (0.733 + 0.679i)45-s + i·49-s + (−0.805 + 0.805i)53-s + 1.89·61-s + (0.0418 + 1.10i)65-s + (−1.19 + 1.19i)73-s − 0.999·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.192 + 0.981i$
Motivic weight: \(5\)
Character: $\chi_{80} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.192 + 0.981i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.617059 - 0.749989i\)
\(L(\frac12)\) \(\approx\) \(0.617059 - 0.749989i\)
\(L(\frac{7}{2})\) 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 \( 1 + (38 - 41i)T \)
good3 \( 1 + 243iT^{2} \)
7 \( 1 - 1.68e4iT^{2} \)
11 \( 1 - 1.61e5T^{2} \)
13 \( 1 + (-475 + 475i)T - 3.71e5iT^{2} \)
17 \( 1 + (1.52e3 + 1.52e3i)T + 1.41e6iT^{2} \)
19 \( 1 + 2.47e6T^{2} \)
23 \( 1 + 6.43e6iT^{2} \)
29 \( 1 + 8.56e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.86e7T^{2} \)
37 \( 1 + (475 + 475i)T + 6.93e7iT^{2} \)
41 \( 1 + 4.95e3T + 1.15e8T^{2} \)
43 \( 1 + 1.47e8iT^{2} \)
47 \( 1 - 2.29e8iT^{2} \)
53 \( 1 + (1.64e4 - 1.64e4i)T - 4.18e8iT^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 - 5.49e4T + 8.44e8T^{2} \)
67 \( 1 - 1.35e9iT^{2} \)
71 \( 1 - 1.80e9T^{2} \)
73 \( 1 + (5.44e4 - 5.44e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 3.07e9T^{2} \)
83 \( 1 + 3.93e9iT^{2} \)
89 \( 1 + 1.40e5iT - 5.58e9T^{2} \)
97 \( 1 + (1.26e5 + 1.26e5i)T + 8.58e9iT^{2} \)
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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

−13.12907050789952461624854822074, −11.81486564221855946187152761869, −11.09472525616520278457004079384, −9.780663943066431139120679917369, −8.496360791828485765075935561412, −7.20041439609260595093301767396, −6.11406684518511628716920339591, −4.19808702174283378467146342239, −2.87274957828654952536962508429, −0.41643144844483045647492667854, 1.69043278917849964065272530981, 3.89100766632856315079018549205, 5.08622599488707583938824691728, 6.75290159913160538177572989232, 8.204744654165001960618709216293, 8.925660761667735830555971901500, 10.62519063083389831877286470180, 11.45810039051531883711777955023, 12.75447727613754868588287289644, 13.50841718049237659912873679449

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