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Properties

Label 2-80-16.5-c5-0-29
Degree $2$
Conductor $80$
Sign $-0.799 + 0.601i$
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  + (−4.24 − 3.73i)2-s + (21.1 − 21.1i)3-s + (4.02 + 31.7i)4-s + (17.6 + 17.6i)5-s + (−168. + 10.6i)6-s + 100. i·7-s + (101. − 149. i)8-s − 651. i·9-s + (−8.91 − 141. i)10-s + (−419. − 419. i)11-s + (756. + 586. i)12-s + (455. − 455. i)13-s + (374. − 425. i)14-s + 747.·15-s + (−991. + 255. i)16-s − 662.·17-s + ⋯
L(s)  = 1  + (−0.750 − 0.661i)2-s + (1.35 − 1.35i)3-s + (0.125 + 0.992i)4-s + (0.316 + 0.316i)5-s + (−1.91 + 0.120i)6-s + 0.772i·7-s + (0.561 − 0.827i)8-s − 2.68i·9-s + (−0.0281 − 0.446i)10-s + (−1.04 − 1.04i)11-s + (1.51 + 1.17i)12-s + (0.748 − 0.748i)13-s + (0.510 − 0.579i)14-s + 0.857·15-s + (−0.968 + 0.249i)16-s − 0.556·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.601i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.799 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.799 + 0.601i$
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: \((2,\ 80,\ (\ :5/2),\ -0.799 + 0.601i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.581090 - 1.73888i\)
\(L(\frac12)\) \(\approx\) \(0.581090 - 1.73888i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.24 + 3.73i)T \)
5 \( 1 + (-17.6 - 17.6i)T \)
good3 \( 1 + (-21.1 + 21.1i)T - 243iT^{2} \)
7 \( 1 - 100. iT - 1.68e4T^{2} \)
11 \( 1 + (419. + 419. i)T + 1.61e5iT^{2} \)
13 \( 1 + (-455. + 455. i)T - 3.71e5iT^{2} \)
17 \( 1 + 662.T + 1.41e6T^{2} \)
19 \( 1 + (-1.20e3 + 1.20e3i)T - 2.47e6iT^{2} \)
23 \( 1 + 2.57e3iT - 6.43e6T^{2} \)
29 \( 1 + (2.02e3 - 2.02e3i)T - 2.05e7iT^{2} \)
31 \( 1 - 3.15e3T + 2.86e7T^{2} \)
37 \( 1 + (-2.28e3 - 2.28e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 3.83e3iT - 1.15e8T^{2} \)
43 \( 1 + (-1.29e4 - 1.29e4i)T + 1.47e8iT^{2} \)
47 \( 1 + 1.56e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.47e4 - 1.47e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-2.44e4 - 2.44e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (1.51e4 - 1.51e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (2.49e4 - 2.49e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 5.20e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.74e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.87e4T + 3.07e9T^{2} \)
83 \( 1 + (-6.19e4 + 6.19e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 1.92e3iT - 5.58e9T^{2} \)
97 \( 1 + 5.72e4T + 8.58e9T^{2} \)
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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

−13.21655367299602929345310027790, −12.02499789489448281315535152561, −10.75277465752535625602243827560, −9.194093876013210324438222515459, −8.468989910381501747594125135409, −7.63036415557974092230952595381, −6.25948052218767547951631636957, −3.09862094191743770586521023283, −2.48464457215236053690351930790, −0.864292880544197253601954676085, 2.02796673507551678166480948605, 4.04266131381691631909437093687, 5.24825912240820820605546213126, 7.37315306113997382068386412969, 8.303416937644730773430438274293, 9.441168718054340367155844135011, 9.999257087155689252198419377456, 10.96760702376147444102386158983, 13.46295869590608234501593641544, 14.02322251742494650013226119803

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