Properties

Degree $2$
Conductor $80$
Sign $-0.538 - 0.842i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−17.2 + 17.2i)3-s + (46.1 − 31.4i)5-s + (154. + 154. i)7-s − 355. i·9-s − 127. i·11-s + (335. + 335. i)13-s + (−254. + 1.34e3i)15-s + (−1.15e3 + 1.15e3i)17-s + 28.2·19-s − 5.32e3·21-s + (−2.78e3 + 2.78e3i)23-s + (1.14e3 − 2.90e3i)25-s + (1.93e3 + 1.93e3i)27-s + 3.38e3i·29-s + 5.38e3i·31-s + ⋯
L(s)  = 1  + (−1.10 + 1.10i)3-s + (0.826 − 0.563i)5-s + (1.18 + 1.18i)7-s − 1.46i·9-s − 0.317i·11-s + (0.549 + 0.549i)13-s + (−0.291 + 1.54i)15-s + (−0.969 + 0.969i)17-s + 0.0179·19-s − 2.63·21-s + (−1.09 + 1.09i)23-s + (0.365 − 0.930i)25-s + (0.511 + 0.511i)27-s + 0.748i·29-s + 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.538 - 0.842i$
Motivic weight: \(5\)
Character: $\chi_{80} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.538 - 0.842i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.648113 + 1.18280i\)
\(L(\frac12)\) \(\approx\) \(0.648113 + 1.18280i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-46.1 + 31.4i)T \)
good3 \( 1 + (17.2 - 17.2i)T - 243iT^{2} \)
7 \( 1 + (-154. - 154. i)T + 1.68e4iT^{2} \)
11 \( 1 + 127. iT - 1.61e5T^{2} \)
13 \( 1 + (-335. - 335. i)T + 3.71e5iT^{2} \)
17 \( 1 + (1.15e3 - 1.15e3i)T - 1.41e6iT^{2} \)
19 \( 1 - 28.2T + 2.47e6T^{2} \)
23 \( 1 + (2.78e3 - 2.78e3i)T - 6.43e6iT^{2} \)
29 \( 1 - 3.38e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.38e3iT - 2.86e7T^{2} \)
37 \( 1 + (-1.15e4 + 1.15e4i)T - 6.93e7iT^{2} \)
41 \( 1 + 1.11e4T + 1.15e8T^{2} \)
43 \( 1 + (1.43e3 - 1.43e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (219. + 219. i)T + 2.29e8iT^{2} \)
53 \( 1 + (-2.27e4 - 2.27e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 2.21e4T + 7.14e8T^{2} \)
61 \( 1 + 1.43e3T + 8.44e8T^{2} \)
67 \( 1 + (2.89e4 + 2.89e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 2.41e4iT - 1.80e9T^{2} \)
73 \( 1 + (-2.85e4 - 2.85e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 2.34e4T + 3.07e9T^{2} \)
83 \( 1 + (1.89e4 - 1.89e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 8.17e3iT - 5.58e9T^{2} \)
97 \( 1 + (7.67e4 - 7.67e4i)T - 8.58e9iT^{2} \)
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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

−13.86546468717099267551836619530, −12.38915275406754223061457553099, −11.43459107612940457712938882447, −10.64679811258606967099234933429, −9.330618433396649484462194459873, −8.505119593413207061302275008301, −6.07854974519861911626100507875, −5.40358819602209873644399799169, −4.31601667154046426009969762375, −1.74226003942223379744404240138, 0.67867472357543196246462170477, 2.02105950092773251258036991155, 4.66937192257155329366336463278, 6.07443470775440321844685602781, 7.00120872240638327161627883000, 8.028095117525120804485390087811, 10.09596799008410886049133413113, 11.03382169624098223246388205384, 11.75182545732196773005342482578, 13.29090827373364627919417530508

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