Properties

Degree $2$
Conductor $80$
Sign $0.448 - 0.893i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.68 − 9.68i)3-s + (−49.1 − 26.5i)5-s + (−48.6 + 48.6i)7-s − 55.4i·9-s + 463. i·11-s + (320. − 320. i)13-s + (219. + 733. i)15-s + (1.04e3 + 1.04e3i)17-s − 701.·19-s + 942.·21-s + (2.00e3 + 2.00e3i)23-s + (1.71e3 + 2.61e3i)25-s + (−2.89e3 + 2.89e3i)27-s − 3.56e3i·29-s + 9.04e3i·31-s + ⋯
L(s)  = 1  + (−0.621 − 0.621i)3-s + (−0.879 − 0.475i)5-s + (−0.375 + 0.375i)7-s − 0.228i·9-s + 1.15i·11-s + (0.526 − 0.526i)13-s + (0.251 + 0.841i)15-s + (0.877 + 0.877i)17-s − 0.445·19-s + 0.466·21-s + (0.789 + 0.789i)23-s + (0.548 + 0.836i)25-s + (−0.762 + 0.762i)27-s − 0.787i·29-s + 1.69i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.448 - 0.893i$
Motivic weight: \(5\)
Character: $\chi_{80} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 0.448 - 0.893i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.605351 + 0.373512i\)
\(L(\frac12)\) \(\approx\) \(0.605351 + 0.373512i\)
\(L(\frac{7}{2})\) 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 \( 1 + (49.1 + 26.5i)T \)
good3 \( 1 + (9.68 + 9.68i)T + 243iT^{2} \)
7 \( 1 + (48.6 - 48.6i)T - 1.68e4iT^{2} \)
11 \( 1 - 463. iT - 1.61e5T^{2} \)
13 \( 1 + (-320. + 320. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-1.04e3 - 1.04e3i)T + 1.41e6iT^{2} \)
19 \( 1 + 701.T + 2.47e6T^{2} \)
23 \( 1 + (-2.00e3 - 2.00e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 3.56e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.04e3iT - 2.86e7T^{2} \)
37 \( 1 + (-1.64e3 - 1.64e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.43e4T + 1.15e8T^{2} \)
43 \( 1 + (3.94e3 + 3.94e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-7.94e3 + 7.94e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (-1.16e4 + 1.16e4i)T - 4.18e8iT^{2} \)
59 \( 1 + 1.12e3T + 7.14e8T^{2} \)
61 \( 1 + 2.93e4T + 8.44e8T^{2} \)
67 \( 1 + (9.19e3 - 9.19e3i)T - 1.35e9iT^{2} \)
71 \( 1 + 5.26e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.79e4 - 2.79e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 8.22e4T + 3.07e9T^{2} \)
83 \( 1 + (-7.72e4 - 7.72e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.45e5iT - 5.58e9T^{2} \)
97 \( 1 + (-9.78e4 - 9.78e4i)T + 8.58e9iT^{2} \)
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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

−13.14633491857512244159881992363, −12.39826804788701776014263429534, −11.80424468542345968822594771583, −10.41179048980873477983091037720, −8.981414295530337157920059228510, −7.72765299845493955922843130673, −6.59134912984299217019755657488, −5.24247902948163695848532739532, −3.58683508466135063380423907859, −1.25480048903888908127183800244, 0.37522031494584147842910580944, 3.21924722325026178164662223178, 4.49361256639331849852771131290, 6.00229402962894823282771388938, 7.34450354559812604215727815640, 8.666162144396038815892064940788, 10.19242980793070151516109387264, 11.07024247383663761639081868267, 11.74083115476170051462862030937, 13.27137209659981804980346608682

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