Properties

Degree $2$
Conductor $80$
Sign $-0.918 + 0.395i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.48 − 7.48i)3-s + (34.7 − 43.8i)5-s + (−19.2 + 19.2i)7-s − 131. i·9-s + 180. i·11-s + (44.2 − 44.2i)13-s + (−587. + 67.9i)15-s + (−621. − 621. i)17-s − 2.67e3·19-s + 287.·21-s + (−2.23e3 − 2.23e3i)23-s + (−712. − 3.04e3i)25-s + (−2.79e3 + 2.79e3i)27-s − 705. i·29-s + 2.76e3i·31-s + ⋯
L(s)  = 1  + (−0.480 − 0.480i)3-s + (0.621 − 0.783i)5-s + (−0.148 + 0.148i)7-s − 0.539i·9-s + 0.450i·11-s + (0.0725 − 0.0725i)13-s + (−0.674 + 0.0779i)15-s + (−0.521 − 0.521i)17-s − 1.69·19-s + 0.142·21-s + (−0.880 − 0.880i)23-s + (−0.228 − 0.973i)25-s + (−0.738 + 0.738i)27-s − 0.155i·29-s + 0.516i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.918 + 0.395i$
Motivic weight: \(5\)
Character: $\chi_{80} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.918 + 0.395i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.185648 - 0.900081i\)
\(L(\frac12)\) \(\approx\) \(0.185648 - 0.900081i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-34.7 + 43.8i)T \)
good3 \( 1 + (7.48 + 7.48i)T + 243iT^{2} \)
7 \( 1 + (19.2 - 19.2i)T - 1.68e4iT^{2} \)
11 \( 1 - 180. iT - 1.61e5T^{2} \)
13 \( 1 + (-44.2 + 44.2i)T - 3.71e5iT^{2} \)
17 \( 1 + (621. + 621. i)T + 1.41e6iT^{2} \)
19 \( 1 + 2.67e3T + 2.47e6T^{2} \)
23 \( 1 + (2.23e3 + 2.23e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 705. iT - 2.05e7T^{2} \)
31 \( 1 - 2.76e3iT - 2.86e7T^{2} \)
37 \( 1 + (3.54e3 + 3.54e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.09e4T + 1.15e8T^{2} \)
43 \( 1 + (5.34e3 + 5.34e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-1.32e4 + 1.32e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (1.56e4 - 1.56e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 4.59e4T + 7.14e8T^{2} \)
61 \( 1 + 1.75e4T + 8.44e8T^{2} \)
67 \( 1 + (-3.10e4 + 3.10e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 1.06e4iT - 1.80e9T^{2} \)
73 \( 1 + (-5.09e4 + 5.09e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 2.47e4T + 3.07e9T^{2} \)
83 \( 1 + (4.89e4 + 4.89e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 2.61e4iT - 5.58e9T^{2} \)
97 \( 1 + (-3.99e4 - 3.99e4i)T + 8.58e9iT^{2} \)
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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

−12.65908039940480201936565805595, −12.24813086326390261440691093032, −10.75481187271266998048408458589, −9.484052425060715621296486749395, −8.481244384268403838663912683460, −6.79422534177500689430566618522, −5.84477399037664692774621219795, −4.37438440634257755749509176793, −2.06616571229224580739448636445, −0.39061172290722094274041907018, 2.18052429341612950612465615346, 4.01582528470056271739568467321, 5.62173652057945035713717491240, 6.63045440539900530615547518668, 8.203084983500022046479385705266, 9.722425993199205011482825074467, 10.65343543907011375453309598564, 11.31925719912318351091439777819, 12.96011722767991110197591705185, 13.86138263444926483173789611741

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