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Properties

Label 2-80-80.29-c5-0-26
Degree $2$
Conductor $80$
Sign $0.327 - 0.944i$
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  + (−5.64 + 0.300i)2-s + (−16.1 + 16.1i)3-s + (31.8 − 3.39i)4-s + (33.0 + 45.0i)5-s + (86.2 − 95.9i)6-s + 106.·7-s + (−178. + 28.6i)8-s − 276. i·9-s + (−200. − 244. i)10-s + (507. − 507. i)11-s + (−458. + 567. i)12-s + (478. − 478. i)13-s + (−602. + 32.0i)14-s + (−1.25e3 − 194. i)15-s + (1.00e3 − 215. i)16-s + 345. i·17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0530i)2-s + (−1.03 + 1.03i)3-s + (0.994 − 0.105i)4-s + (0.591 + 0.806i)5-s + (0.978 − 1.08i)6-s + 0.822·7-s + (−0.987 + 0.158i)8-s − 1.13i·9-s + (−0.633 − 0.774i)10-s + (1.26 − 1.26i)11-s + (−0.918 + 1.13i)12-s + (0.784 − 0.784i)13-s + (−0.821 + 0.0436i)14-s + (−1.44 − 0.222i)15-s + (0.977 − 0.210i)16-s + 0.289i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.327 - 0.944i$
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: \((2,\ 80,\ (\ :5/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.901514 + 0.641792i\)
\(L(\frac12)\) \(\approx\) \(0.901514 + 0.641792i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.64 - 0.300i)T \)
5 \( 1 + (-33.0 - 45.0i)T \)
good3 \( 1 + (16.1 - 16.1i)T - 243iT^{2} \)
7 \( 1 - 106.T + 1.68e4T^{2} \)
11 \( 1 + (-507. + 507. i)T - 1.61e5iT^{2} \)
13 \( 1 + (-478. + 478. i)T - 3.71e5iT^{2} \)
17 \( 1 - 345. iT - 1.41e6T^{2} \)
19 \( 1 + (-1.70e3 - 1.70e3i)T + 2.47e6iT^{2} \)
23 \( 1 - 1.76e3T + 6.43e6T^{2} \)
29 \( 1 + (3.73e3 + 3.73e3i)T + 2.05e7iT^{2} \)
31 \( 1 - 9.85e3T + 2.86e7T^{2} \)
37 \( 1 + (4.21e3 + 4.21e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 2.72e3iT - 1.15e8T^{2} \)
43 \( 1 + (1.66e3 + 1.66e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 9.10e3iT - 2.29e8T^{2} \)
53 \( 1 + (9.65e3 + 9.65e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (2.37e4 - 2.37e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (-2.02e4 - 2.02e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (-3.56e4 + 3.56e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 2.91e4iT - 1.80e9T^{2} \)
73 \( 1 - 1.15e4T + 2.07e9T^{2} \)
79 \( 1 + 4.11e4T + 3.07e9T^{2} \)
83 \( 1 + (1.77e4 - 1.77e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 9.04e4iT - 5.58e9T^{2} \)
97 \( 1 - 6.52e4iT - 8.58e9T^{2} \)
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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

−13.94094671377856109939428692102, −11.73255385045244344996354006415, −11.16164928688121104410301977733, −10.41161944511285586978757461650, −9.420171996115818506223978095395, −8.076348769840495230147760857095, −6.31691674990505798820435003869, −5.62034286569618763126263269952, −3.47003042950345881157092666409, −1.13112021010587339775037283085, 1.04197158760087733003806889197, 1.70548670314949443121461930086, 4.96523773869639820314574978899, 6.44810267055180514839321835852, 7.21795741950950150454114772238, 8.722531355803458980389416558606, 9.682868966172838168136641015093, 11.35724259080597952690364780781, 11.78981363416474248488993692511, 12.81538589528738808408100635112

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