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Properties

Label 2-80-16.5-c5-0-1
Degree $2$
Conductor $80$
Sign $-0.929 + 0.368i$
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  + (−0.0219 + 5.65i)2-s + (9.88 − 9.88i)3-s + (−31.9 − 0.248i)4-s + (−17.6 − 17.6i)5-s + (55.7 + 56.1i)6-s + 122. i·7-s + (2.11 − 181. i)8-s + 47.4i·9-s + (100. − 99.6i)10-s + (−301. − 301. i)11-s + (−318. + 313. i)12-s + (−384. + 384. i)13-s + (−692. − 2.68i)14-s − 349.·15-s + (1.02e3 + 15.9i)16-s − 1.55e3·17-s + ⋯
L(s)  = 1  + (−0.00388 + 0.999i)2-s + (0.634 − 0.634i)3-s + (−0.999 − 0.00777i)4-s + (−0.316 − 0.316i)5-s + (0.631 + 0.636i)6-s + 0.943i·7-s + (0.0116 − 0.999i)8-s + 0.195i·9-s + (0.317 − 0.314i)10-s + (−0.750 − 0.750i)11-s + (−0.639 + 0.629i)12-s + (−0.630 + 0.630i)13-s + (−0.943 − 0.00366i)14-s − 0.401·15-s + (0.999 + 0.0155i)16-s − 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.929 + 0.368i$
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: \((2,\ 80,\ (\ :5/2),\ -0.929 + 0.368i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0702595 - 0.368152i\)
\(L(\frac12)\) \(\approx\) \(0.0702595 - 0.368152i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0219 - 5.65i)T \)
5 \( 1 + (17.6 + 17.6i)T \)
good3 \( 1 + (-9.88 + 9.88i)T - 243iT^{2} \)
7 \( 1 - 122. iT - 1.68e4T^{2} \)
11 \( 1 + (301. + 301. i)T + 1.61e5iT^{2} \)
13 \( 1 + (384. - 384. i)T - 3.71e5iT^{2} \)
17 \( 1 + 1.55e3T + 1.41e6T^{2} \)
19 \( 1 + (2.09e3 - 2.09e3i)T - 2.47e6iT^{2} \)
23 \( 1 + 4.76e3iT - 6.43e6T^{2} \)
29 \( 1 + (1.90e3 - 1.90e3i)T - 2.05e7iT^{2} \)
31 \( 1 - 1.15e3T + 2.86e7T^{2} \)
37 \( 1 + (7.42e3 + 7.42e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 4.11e3iT - 1.15e8T^{2} \)
43 \( 1 + (-8.40e3 - 8.40e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 8.94e3T + 2.29e8T^{2} \)
53 \( 1 + (-2.49e4 - 2.49e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-2.32e4 - 2.32e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-1.43e4 + 1.43e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (3.88e4 - 3.88e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 1.28e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.85e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.07e4T + 3.07e9T^{2} \)
83 \( 1 + (3.94e4 - 3.94e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 9.72e4iT - 5.58e9T^{2} \)
97 \( 1 + 4.67e4T + 8.58e9T^{2} \)
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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

−14.18253296032592392060296614263, −13.03818648593747435370601568019, −12.38420953962124997438865316628, −10.57211214001671383703287813069, −8.701459970873210232643134152939, −8.521906292234882733247004965188, −7.17848190030762508011995662879, −5.88308433333151282027420365418, −4.44944601748151725554326069846, −2.33384681503269718540664412331, 0.13902705155782104261198314950, 2.44292537434020180531653086812, 3.78095140246981809765475335948, 4.79254874219465121702444316980, 7.17691942612491507536364209501, 8.538179408489717044469338070157, 9.723304861311360364058529743561, 10.47909553576614327666987851424, 11.48729676831953169460007141287, 12.93054407927768434604203622982

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