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Properties

Label 2-80-16.3-c2-0-4
Degree $2$
Conductor $80$
Sign $-0.556 - 0.830i$
Analytic cond. $2.17984$
Root an. cond. $1.47642$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.71 + 1.02i)2-s + (−3.48 + 3.48i)3-s + (1.88 + 3.52i)4-s + (1.58 − 1.58i)5-s + (−9.55 + 2.39i)6-s − 6.27·7-s + (−0.384 + 7.99i)8-s − 15.2i·9-s + (4.33 − 1.08i)10-s + (10.5 + 10.5i)11-s + (−18.8 − 5.70i)12-s + (4.99 + 4.99i)13-s + (−10.7 − 6.44i)14-s + 11.0i·15-s + (−8.87 + 13.3i)16-s + 16.5·17-s + ⋯
L(s)  = 1  + (0.857 + 0.513i)2-s + (−1.16 + 1.16i)3-s + (0.471 + 0.881i)4-s + (0.316 − 0.316i)5-s + (−1.59 + 0.399i)6-s − 0.895·7-s + (−0.0481 + 0.998i)8-s − 1.69i·9-s + (0.433 − 0.108i)10-s + (0.955 + 0.955i)11-s + (−1.57 − 0.475i)12-s + (0.384 + 0.384i)13-s + (−0.768 − 0.460i)14-s + 0.734i·15-s + (−0.554 + 0.832i)16-s + 0.973·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.556 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.556 - 0.830i$
Analytic conductor: \(2.17984\)
Root analytic conductor: \(1.47642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :1),\ -0.556 - 0.830i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.665339 + 1.24667i\)
\(L(\frac12)\) \(\approx\) \(0.665339 + 1.24667i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.71 - 1.02i)T \)
5 \( 1 + (-1.58 + 1.58i)T \)
good3 \( 1 + (3.48 - 3.48i)T - 9iT^{2} \)
7 \( 1 + 6.27T + 49T^{2} \)
11 \( 1 + (-10.5 - 10.5i)T + 121iT^{2} \)
13 \( 1 + (-4.99 - 4.99i)T + 169iT^{2} \)
17 \( 1 - 16.5T + 289T^{2} \)
19 \( 1 + (-25.9 + 25.9i)T - 361iT^{2} \)
23 \( 1 + 21.6T + 529T^{2} \)
29 \( 1 + (4.57 + 4.57i)T + 841iT^{2} \)
31 \( 1 - 9.09iT - 961T^{2} \)
37 \( 1 + (12.8 - 12.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 23.8iT - 1.68e3T^{2} \)
43 \( 1 + (-36.3 - 36.3i)T + 1.84e3iT^{2} \)
47 \( 1 + 91.0iT - 2.20e3T^{2} \)
53 \( 1 + (-37.1 + 37.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (27.3 + 27.3i)T + 3.48e3iT^{2} \)
61 \( 1 + (-11.6 - 11.6i)T + 3.72e3iT^{2} \)
67 \( 1 + (46.0 - 46.0i)T - 4.48e3iT^{2} \)
71 \( 1 + 56.7T + 5.04e3T^{2} \)
73 \( 1 - 28.1iT - 5.32e3T^{2} \)
79 \( 1 + 58.5iT - 6.24e3T^{2} \)
83 \( 1 + (-38.2 + 38.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 19.9iT - 7.92e3T^{2} \)
97 \( 1 - 89.5T + 9.40e3T^{2} \)
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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

−14.68841103312612989860801437872, −13.47366944868923598549547723509, −12.17059951055456165054743913500, −11.59950261103620539703381810918, −10.09084494488754795992470872904, −9.201538769023531240495072265725, −7.00156144789955211907293892345, −5.93292725359708503489798399898, −4.87451871591072758708656716653, −3.68539422998520539851354828273, 1.19934622450119079280027569920, 3.41672180415358152250975415395, 5.87750245590328310138230036232, 6.02053927475963947452045520811, 7.44657751488836981271395124508, 9.742576798346894005136506701016, 10.88448064240127412006883631668, 11.93118711247252581244991779331, 12.47892305933910090699317094471, 13.62223125338574807070283033769

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