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Properties

Label 2-80-80.29-c5-0-13
Degree $2$
Conductor $80$
Sign $0.530 - 0.847i$
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  + (−4.71 − 3.13i)2-s + (−21.0 + 21.0i)3-s + (12.4 + 29.4i)4-s + (9.85 − 55.0i)5-s + (164. − 33.2i)6-s − 32.3·7-s + (33.9 − 177. i)8-s − 639. i·9-s + (−218. + 228. i)10-s + (−230. + 230. i)11-s + (−880. − 359. i)12-s + (651. − 651. i)13-s + (152. + 101. i)14-s + (948. + 1.36e3i)15-s + (−716. + 731. i)16-s + 1.17e3i·17-s + ⋯
L(s)  = 1  + (−0.832 − 0.553i)2-s + (−1.34 + 1.34i)3-s + (0.387 + 0.921i)4-s + (0.176 − 0.984i)5-s + (1.86 − 0.376i)6-s − 0.249·7-s + (0.187 − 0.982i)8-s − 2.63i·9-s + (−0.691 + 0.722i)10-s + (−0.575 + 0.575i)11-s + (−1.76 − 0.720i)12-s + (1.06 − 1.06i)13-s + (0.207 + 0.137i)14-s + (1.08 + 1.56i)15-s + (−0.699 + 0.714i)16-s + 0.987i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.530 - 0.847i$
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: \((2,\ 80,\ (\ :5/2),\ 0.530 - 0.847i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.480202 + 0.265913i\)
\(L(\frac12)\) \(\approx\) \(0.480202 + 0.265913i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.71 + 3.13i)T \)
5 \( 1 + (-9.85 + 55.0i)T \)
good3 \( 1 + (21.0 - 21.0i)T - 243iT^{2} \)
7 \( 1 + 32.3T + 1.68e4T^{2} \)
11 \( 1 + (230. - 230. i)T - 1.61e5iT^{2} \)
13 \( 1 + (-651. + 651. i)T - 3.71e5iT^{2} \)
17 \( 1 - 1.17e3iT - 1.41e6T^{2} \)
19 \( 1 + (242. + 242. i)T + 2.47e6iT^{2} \)
23 \( 1 + 48.1T + 6.43e6T^{2} \)
29 \( 1 + (-5.17e3 - 5.17e3i)T + 2.05e7iT^{2} \)
31 \( 1 + 3.32e3T + 2.86e7T^{2} \)
37 \( 1 + (1.79e3 + 1.79e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 9.79e3iT - 1.15e8T^{2} \)
43 \( 1 + (2.56e3 + 2.56e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 2.02e4iT - 2.29e8T^{2} \)
53 \( 1 + (-1.72e4 - 1.72e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-1.52e4 + 1.52e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (-1.99e4 - 1.99e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (-4.13e4 + 4.13e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 1.93e4iT - 1.80e9T^{2} \)
73 \( 1 - 3.97e4T + 2.07e9T^{2} \)
79 \( 1 - 8.64e4T + 3.07e9T^{2} \)
83 \( 1 + (1.42e4 - 1.42e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 8.09e4iT - 5.58e9T^{2} \)
97 \( 1 - 8.80e4iT - 8.58e9T^{2} \)
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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

−12.89804219394925931219680876258, −12.32244671725971914740565584616, −10.99383589941821062533250253714, −10.37675861078745671526381330842, −9.441904301246048445747123924390, −8.336358775391727633975694024545, −6.26372032206262670780122970223, −4.98859166755172062082290188590, −3.66530255027311399973624567998, −0.915795318887771862629111902314, 0.50437854963406428162393397018, 2.11146006435059274600003988883, 5.49400002322229909208490433847, 6.47176707923960647419833441322, 7.05386988087122585377854973548, 8.298795535641374812996810593196, 10.12603494845478621918452911918, 11.16327293333627855438596420160, 11.69259644616882130112480250189, 13.41467481440486280222617522924

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