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Properties

Label 2-80-16.11-c10-0-5
Degree $2$
Conductor $80$
Sign $0.428 - 0.903i$
Analytic cond. $50.8285$
Root an. cond. $7.12941$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (26.8 − 17.4i)2-s + (−61.0 − 61.0i)3-s + (415. − 935. i)4-s + (−988. − 988. i)5-s + (−2.70e3 − 573. i)6-s − 2.17e4·7-s + (−5.17e3 − 3.23e4i)8-s − 5.15e4i·9-s + (−4.37e4 − 9.27e3i)10-s + (−1.84e5 + 1.84e5i)11-s + (−8.25e4 + 3.17e4i)12-s + (1.63e5 − 1.63e5i)13-s + (−5.84e5 + 3.80e5i)14-s + 1.20e5i·15-s + (−7.03e5 − 7.77e5i)16-s + 1.37e6·17-s + ⋯
L(s)  = 1  + (0.838 − 0.545i)2-s + (−0.251 − 0.251i)3-s + (0.405 − 0.913i)4-s + (−0.316 − 0.316i)5-s + (−0.347 − 0.0737i)6-s − 1.29·7-s + (−0.157 − 0.987i)8-s − 0.873i·9-s + (−0.437 − 0.0927i)10-s + (−1.14 + 1.14i)11-s + (−0.331 + 0.127i)12-s + (0.439 − 0.439i)13-s + (−1.08 + 0.706i)14-s + 0.158i·15-s + (−0.670 − 0.741i)16-s + 0.969·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $0.428 - 0.903i$
Analytic conductor: \(50.8285\)
Root analytic conductor: \(7.12941\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 80,\ (\ :5),\ 0.428 - 0.903i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.334495 + 0.211492i\)
\(L(\frac12)\) \(\approx\) \(0.334495 + 0.211492i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-26.8 + 17.4i)T \)
5 \( 1 + (988. + 988. i)T \)
good3 \( 1 + (61.0 + 61.0i)T + 5.90e4iT^{2} \)
7 \( 1 + 2.17e4T + 2.82e8T^{2} \)
11 \( 1 + (1.84e5 - 1.84e5i)T - 2.59e10iT^{2} \)
13 \( 1 + (-1.63e5 + 1.63e5i)T - 1.37e11iT^{2} \)
17 \( 1 - 1.37e6T + 2.01e12T^{2} \)
19 \( 1 + (-2.80e6 - 2.80e6i)T + 6.13e12iT^{2} \)
23 \( 1 + 1.03e7T + 4.14e13T^{2} \)
29 \( 1 + (-2.69e7 + 2.69e7i)T - 4.20e14iT^{2} \)
31 \( 1 - 3.26e7iT - 8.19e14T^{2} \)
37 \( 1 + (3.57e7 + 3.57e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 1.14e8iT - 1.34e16T^{2} \)
43 \( 1 + (1.12e7 - 1.12e7i)T - 2.16e16iT^{2} \)
47 \( 1 - 1.82e8iT - 5.25e16T^{2} \)
53 \( 1 + (-4.17e7 - 4.17e7i)T + 1.74e17iT^{2} \)
59 \( 1 + (-4.21e8 + 4.21e8i)T - 5.11e17iT^{2} \)
61 \( 1 + (5.26e8 - 5.26e8i)T - 7.13e17iT^{2} \)
67 \( 1 + (-6.08e8 - 6.08e8i)T + 1.82e18iT^{2} \)
71 \( 1 + 8.02e8T + 3.25e18T^{2} \)
73 \( 1 - 2.53e9iT - 4.29e18T^{2} \)
79 \( 1 + 3.51e9iT - 9.46e18T^{2} \)
83 \( 1 + (8.61e8 + 8.61e8i)T + 1.55e19iT^{2} \)
89 \( 1 - 6.51e9iT - 3.11e19T^{2} \)
97 \( 1 + 9.80e9T + 7.37e19T^{2} \)
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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

−12.32611381664972941550003571619, −12.05086156707444474281048693426, −10.17299527741605827926328557582, −9.776785371540863514286997968674, −7.74868486181940579712926354393, −6.39978513657847883398556082659, −5.46074825185948140070258332428, −3.90504605221692727332858155645, −2.90241874973285353112522739858, −1.18053027162428868561273747648, 0.084624354598470105075702451888, 2.71863534271209826978744855784, 3.59333281302126019329681311818, 5.16075633367834551565511438725, 6.09732376124715276871885291478, 7.33950586718503970915140466128, 8.421893273174817878368540345807, 10.12502231201364648515978957682, 11.18365137372303275393756349495, 12.26783612588673036595423142978

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