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Properties

Label 2-80-80.29-c5-0-54
Degree $2$
Conductor $80$
Sign $-0.969 - 0.243i$
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  + (−5.61 − 0.676i)2-s + (18.5 − 18.5i)3-s + (31.0 + 7.60i)4-s + (8.85 − 55.1i)5-s + (−116. + 91.6i)6-s − 206.·7-s + (−169. − 63.7i)8-s − 445. i·9-s + (−87.0 + 303. i)10-s + (−431. + 431. i)11-s + (717. − 435. i)12-s + (325. − 325. i)13-s + (1.15e3 + 139. i)14-s + (−859. − 1.18e3i)15-s + (908. + 472. i)16-s + 11.3i·17-s + ⋯
L(s)  = 1  + (−0.992 − 0.119i)2-s + (1.19 − 1.19i)3-s + (0.971 + 0.237i)4-s + (0.158 − 0.987i)5-s + (−1.32 + 1.03i)6-s − 1.59·7-s + (−0.935 − 0.352i)8-s − 1.83i·9-s + (−0.275 + 0.961i)10-s + (−1.07 + 1.07i)11-s + (1.43 − 0.873i)12-s + (0.533 − 0.533i)13-s + (1.58 + 0.190i)14-s + (−0.986 − 1.36i)15-s + (0.887 + 0.461i)16-s + 0.00949i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.969 - 0.243i$
Analytic conductor: \(12.8307\)
Root analytic conductor: \(3.58199\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5/2),\ -0.969 - 0.243i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.106139 + 0.858375i\)
\(L(\frac12)\) \(\approx\) \(0.106139 + 0.858375i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.61 + 0.676i)T \)
5 \( 1 + (-8.85 + 55.1i)T \)
good3 \( 1 + (-18.5 + 18.5i)T - 243iT^{2} \)
7 \( 1 + 206.T + 1.68e4T^{2} \)
11 \( 1 + (431. - 431. i)T - 1.61e5iT^{2} \)
13 \( 1 + (-325. + 325. i)T - 3.71e5iT^{2} \)
17 \( 1 - 11.3iT - 1.41e6T^{2} \)
19 \( 1 + (-633. - 633. i)T + 2.47e6iT^{2} \)
23 \( 1 - 633.T + 6.43e6T^{2} \)
29 \( 1 + (5.80e3 + 5.80e3i)T + 2.05e7iT^{2} \)
31 \( 1 - 4.46e3T + 2.86e7T^{2} \)
37 \( 1 + (-1.63e3 - 1.63e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 2.09e3iT - 1.15e8T^{2} \)
43 \( 1 + (3.51e3 + 3.51e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 8.22e3iT - 2.29e8T^{2} \)
53 \( 1 + (1.44e4 + 1.44e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-6.02e3 + 6.02e3i)T - 7.14e8iT^{2} \)
61 \( 1 + (2.77e4 + 2.77e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (1.16e4 - 1.16e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 7.39e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.38e4T + 2.07e9T^{2} \)
79 \( 1 - 4.59e3T + 3.07e9T^{2} \)
83 \( 1 + (2.49e4 - 2.49e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.27e5iT - 5.58e9T^{2} \)
97 \( 1 - 1.76e5iT - 8.58e9T^{2} \)
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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

−12.94754287992359009813150354469, −12.12643434219382653717452888854, −10.01731432865608277684188236596, −9.333018200713481737213126828044, −8.202597807824446173448507175873, −7.41006922810288795205919044436, −6.13244080071774064644291607213, −3.21434490136519366198481217678, −1.95509453108907095662274673568, −0.40399325607298335000794711518, 2.78123861910363802959042423294, 3.35569798664657028693693681570, 5.96291499821262486806951410548, 7.32540202639975971484225906511, 8.700722749133058308575498880772, 9.514927882837883330637652608411, 10.34185256914728713969687602322, 11.10867054107347402913829141005, 13.24353633504924491435900032200, 14.23886193207443158054946746290

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