Properties

Degree $2$
Conductor $80$
Sign $0.801 - 0.597i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.801 - 0.597i$
Motivic weight: \(5\)
Character: $\chi_{80} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 0.801 - 0.597i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.20740 + 0.731988i\)
\(L(\frac12)\) \(\approx\) \(2.20740 + 0.731988i\)
\(L(\frac{7}{2})\) 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

−13.66574823006680131063273157222, −12.65408297388201713072497188265, −11.23434454665927710782847008157, −10.22552570460735633915310258560, −8.977779885150276298193767779257, −7.64314495602464499175779129581, −6.65000936873731857965772288326, −5.10609899289802986061832464431, −3.01976329990969455459788390291, −1.74441135995439221018530639549, 1.04881376529821229929926917762, 3.09596306312797431798723083943, 4.66602034276215601846787340793, 5.97433934825477632328688239038, 7.67984286367559694353717808100, 9.159766820078813573103505587327, 9.512417155276654334642829481630, 11.07025643541225067851246663907, 12.28556454312863565751983018468, 13.50683030787270425493907545685

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