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Properties

Label 2-80-16.5-c5-0-21
Degree $2$
Conductor $80$
Sign $0.989 - 0.144i$
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  + (−0.350 + 5.64i)2-s + (−18.9 + 18.9i)3-s + (−31.7 − 3.95i)4-s + (17.6 + 17.6i)5-s + (−100. − 113. i)6-s − 174. i·7-s + (33.4 − 177. i)8-s − 471. i·9-s + (−106. + 93.6i)10-s + (189. + 189. i)11-s + (675. − 525. i)12-s + (−782. + 782. i)13-s + (984. + 61.1i)14-s − 668.·15-s + (992. + 251. i)16-s − 923.·17-s + ⋯
L(s)  = 1  + (−0.0619 + 0.998i)2-s + (−1.21 + 1.21i)3-s + (−0.992 − 0.123i)4-s + (0.316 + 0.316i)5-s + (−1.13 − 1.28i)6-s − 1.34i·7-s + (0.184 − 0.982i)8-s − 1.94i·9-s + (−0.335 + 0.296i)10-s + (0.471 + 0.471i)11-s + (1.35 − 1.05i)12-s + (−1.28 + 1.28i)13-s + (1.34 + 0.0833i)14-s − 0.766·15-s + (0.969 + 0.245i)16-s − 0.775·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.989 - 0.144i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ 0.989 - 0.144i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.517000 + 0.0374889i\)
\(L(\frac12)\) \(\approx\) \(0.517000 + 0.0374889i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.350 - 5.64i)T \)
5 \( 1 + (-17.6 - 17.6i)T \)
good3 \( 1 + (18.9 - 18.9i)T - 243iT^{2} \)
7 \( 1 + 174. iT - 1.68e4T^{2} \)
11 \( 1 + (-189. - 189. i)T + 1.61e5iT^{2} \)
13 \( 1 + (782. - 782. i)T - 3.71e5iT^{2} \)
17 \( 1 + 923.T + 1.41e6T^{2} \)
19 \( 1 + (-898. + 898. i)T - 2.47e6iT^{2} \)
23 \( 1 + 3.82e3iT - 6.43e6T^{2} \)
29 \( 1 + (-978. + 978. i)T - 2.05e7iT^{2} \)
31 \( 1 - 4.97e3T + 2.86e7T^{2} \)
37 \( 1 + (5.78e3 + 5.78e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 639. iT - 1.15e8T^{2} \)
43 \( 1 + (-8.06e3 - 8.06e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 2.20e4T + 2.29e8T^{2} \)
53 \( 1 + (1.43e4 + 1.43e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (2.85e4 + 2.85e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-1.92e4 + 1.92e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (2.51e4 - 2.51e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 5.28e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.07e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.13e4T + 3.07e9T^{2} \)
83 \( 1 + (-1.84e4 + 1.84e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 5.36e4iT - 5.58e9T^{2} \)
97 \( 1 - 5.21e4T + 8.58e9T^{2} \)
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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

−13.88694433212035231665260642118, −12.26343761170635414977495264955, −10.92185085142806215437437685052, −10.01035002042957815853801164308, −9.257941173749779904158524548761, −7.12711095565313000036792052713, −6.42943698950063189955016484291, −4.75144222359760753391354932385, −4.26330649162360310874367320982, −0.30624086694744781439462102426, 1.18434082931138522537258791842, 2.57303642206282020527205412807, 5.19107213760882198057948940420, 5.86134830665009118733264894302, 7.66452627052451370152737148741, 8.993087105041723111048079377138, 10.33719132295422076556439037650, 11.68959426803240985587917236801, 12.11306567541171784288507332507, 12.89857066083997411685796524028

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