Properties

Label 2-80-16.5-c5-0-14
Degree $2$
Conductor $80$
Sign $0.418 - 0.908i$
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  + (−5.55 + 1.04i)2-s + (9.77 − 9.77i)3-s + (29.7 − 11.6i)4-s + (17.6 + 17.6i)5-s + (−44.0 + 64.5i)6-s + 103. i·7-s + (−153. + 96.0i)8-s + 51.9i·9-s + (−116. − 79.7i)10-s + (1.18 + 1.18i)11-s + (177. − 405. i)12-s + (−196. + 196. i)13-s + (−108. − 576. i)14-s + 345.·15-s + (752. − 695. i)16-s + 274.·17-s + ⋯
L(s)  = 1  + (−0.982 + 0.185i)2-s + (0.626 − 0.626i)3-s + (0.931 − 0.364i)4-s + (0.316 + 0.316i)5-s + (−0.499 + 0.732i)6-s + 0.799i·7-s + (−0.847 + 0.530i)8-s + 0.213i·9-s + (−0.369 − 0.252i)10-s + (0.00294 + 0.00294i)11-s + (0.355 − 0.812i)12-s + (−0.322 + 0.322i)13-s + (−0.148 − 0.785i)14-s + 0.396·15-s + (0.734 − 0.678i)16-s + 0.230·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.418 - 0.908i$
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: \((2,\ 80,\ (\ :5/2),\ 0.418 - 0.908i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.11100 + 0.711136i\)
\(L(\frac12)\) \(\approx\) \(1.11100 + 0.711136i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.55 - 1.04i)T \)
5 \( 1 + (-17.6 - 17.6i)T \)
good3 \( 1 + (-9.77 + 9.77i)T - 243iT^{2} \)
7 \( 1 - 103. iT - 1.68e4T^{2} \)
11 \( 1 + (-1.18 - 1.18i)T + 1.61e5iT^{2} \)
13 \( 1 + (196. - 196. i)T - 3.71e5iT^{2} \)
17 \( 1 - 274.T + 1.41e6T^{2} \)
19 \( 1 + (464. - 464. i)T - 2.47e6iT^{2} \)
23 \( 1 - 3.17e3iT - 6.43e6T^{2} \)
29 \( 1 + (3.61e3 - 3.61e3i)T - 2.05e7iT^{2} \)
31 \( 1 - 8.59e3T + 2.86e7T^{2} \)
37 \( 1 + (-5.38e3 - 5.38e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.27e4iT - 1.15e8T^{2} \)
43 \( 1 + (-2.05e3 - 2.05e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 19.2T + 2.29e8T^{2} \)
53 \( 1 + (-2.84e3 - 2.84e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (1.60e4 + 1.60e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-2.82e4 + 2.82e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (4.16e4 - 4.16e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 2.53e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.47e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.18e3T + 3.07e9T^{2} \)
83 \( 1 + (7.37e4 - 7.37e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 3.10e4iT - 5.58e9T^{2} \)
97 \( 1 - 4.83e4T + 8.58e9T^{2} \)
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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

−13.76308042333599549908654253291, −12.42272016256137751981420201790, −11.30694126260605696343759490862, −10.01659361837313948332406221175, −8.963106193680704346173924872365, −7.962916103425261252962505387177, −6.93001151031587881245994430582, −5.59024709818011434079421671582, −2.77615289984208802932849702498, −1.65331777810643451632619566351, 0.72588590877140002971982468468, 2.69119318949375170322656647631, 4.22936530193099722574561355178, 6.36112135806175542114586947839, 7.78175908458725618234287242100, 8.851101357911151366420240616691, 9.829725534811597094625469092616, 10.54406262440389400915633604049, 11.93865386749348658380412432686, 13.16528426640587887858820249535

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