Properties

Degree $2$
Conductor $80$
Sign $0.549 - 0.835i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.549 - 0.835i$
Motivic weight: \(5\)
Character: $\chi_{80} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 0.549 - 0.835i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.57944 + 0.851321i\)
\(L(\frac12)\) \(\approx\) \(1.57944 + 0.851321i\)
\(L(\frac{7}{2})\) 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

−13.77551564960710290125560042682, −12.36005953098642605595706059514, −11.65541850907098675874939440363, −10.29010214691275498677987680626, −8.814434877078632696602391922186, −7.69922766881886781064180361428, −6.99113388313083275719158999920, −4.97956812227033851777444476520, −3.28193103182487545986228732021, −1.72768310764690907379276228452, 0.76124038033039586460059235960, 3.33433528116155110741872532270, 4.23470592923594339350518334738, 5.95651030157787048834987263354, 7.942201512877393631302615980625, 8.473763672092560895917326102446, 9.835522257150010145292328648065, 11.03307152421322364429744937521, 12.07550005687298249910052822898, 13.35267535115370699534100282462

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