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Properties

Label 2-80-80.29-c5-0-46
Degree $2$
Conductor $80$
Sign $-0.549 + 0.835i$
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  + (−2.78 − 4.92i)2-s + (16.5 − 16.5i)3-s + (−16.4 + 27.4i)4-s + (55.5 − 5.97i)5-s + (−127. − 35.4i)6-s + 50.7·7-s + (180. + 4.65i)8-s − 307. i·9-s + (−184. − 256. i)10-s + (214. − 214. i)11-s + (181. + 728. i)12-s + (440. − 440. i)13-s + (−141. − 249. i)14-s + (823. − 1.02e3i)15-s + (−481. − 903. i)16-s + 927. i·17-s + ⋯
L(s)  = 1  + (−0.492 − 0.870i)2-s + (1.06 − 1.06i)3-s + (−0.514 + 0.857i)4-s + (0.994 − 0.106i)5-s + (−1.45 − 0.402i)6-s + 0.391·7-s + (0.999 + 0.0257i)8-s − 1.26i·9-s + (−0.582 − 0.812i)10-s + (0.534 − 0.534i)11-s + (0.364 + 1.46i)12-s + (0.723 − 0.723i)13-s + (−0.192 − 0.340i)14-s + (0.944 − 1.17i)15-s + (−0.470 − 0.882i)16-s + 0.778i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.549 + 0.835i$
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: \((2,\ 80,\ (\ :5/2),\ -0.549 + 0.835i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.11750 - 2.07218i\)
\(L(\frac12)\) \(\approx\) \(1.11750 - 2.07218i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.78 + 4.92i)T \)
5 \( 1 + (-55.5 + 5.97i)T \)
good3 \( 1 + (-16.5 + 16.5i)T - 243iT^{2} \)
7 \( 1 - 50.7T + 1.68e4T^{2} \)
11 \( 1 + (-214. + 214. i)T - 1.61e5iT^{2} \)
13 \( 1 + (-440. + 440. i)T - 3.71e5iT^{2} \)
17 \( 1 - 927. iT - 1.41e6T^{2} \)
19 \( 1 + (-653. - 653. i)T + 2.47e6iT^{2} \)
23 \( 1 + 4.32e3T + 6.43e6T^{2} \)
29 \( 1 + (3.47e3 + 3.47e3i)T + 2.05e7iT^{2} \)
31 \( 1 + 6.98e3T + 2.86e7T^{2} \)
37 \( 1 + (-6.87e3 - 6.87e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 7.97e3iT - 1.15e8T^{2} \)
43 \( 1 + (1.61e3 + 1.61e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 1.53e4iT - 2.29e8T^{2} \)
53 \( 1 + (-1.65e4 - 1.65e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (1.12e4 - 1.12e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (-3.13e4 - 3.13e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (-3.98e4 + 3.98e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 5.65e4iT - 1.80e9T^{2} \)
73 \( 1 + 1.92e4T + 2.07e9T^{2} \)
79 \( 1 + 1.31e4T + 3.07e9T^{2} \)
83 \( 1 + (-5.93e4 + 5.93e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 1.02e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.08e5iT - 8.58e9T^{2} \)
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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

−13.15028717984290675552131265434, −12.13794404683067412829906320509, −10.75354192561640353900334339267, −9.514400561488108131535021954267, −8.501461687968791735717283507973, −7.74022616938423935748511604342, −6.00938522097718653095828598969, −3.61680767937980758286362652107, −2.15020934053999559127756625230, −1.22094244650235350612036078807, 1.89768592812676507780697326210, 4.03324488570725751083844131390, 5.37763095986092535338086627517, 6.89611400242405790141695932222, 8.373292564901177912687144585848, 9.391742334546555308567508338660, 9.780924005906484968854076285980, 11.12732643734440315197119051638, 13.33701971360291073686765255235, 14.38937001743175117856772668841

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