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Properties

Label 2-80-20.7-c5-0-10
Degree $2$
Conductor $80$
Sign $0.549 + 0.835i$
Analytic cond. $12.8307$
Root an. cond. $3.58199$
Motivic weight $5$
Arithmetic yes
Rational no
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: $0.549 + 0.835i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (47, \cdot )$
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

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

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