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Properties

Label 2-160-20.3-c5-0-8
Degree $2$
Conductor $160$
Sign $0.869 - 0.494i$
Analytic cond. $25.6614$
Root an. cond. $5.06570$
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  + (−4.47 + 4.47i)3-s + (−49.0 − 26.8i)5-s + (−132. − 132. i)7-s + 202. i·9-s + 364. i·11-s + (−530. − 530. i)13-s + (339. − 99.5i)15-s + (115. − 115. i)17-s + 1.49e3·19-s + 1.18e3·21-s + (2.63e3 − 2.63e3i)23-s + (1.68e3 + 2.63e3i)25-s + (−1.99e3 − 1.99e3i)27-s + 8.08e3i·29-s − 1.29e3i·31-s + ⋯
L(s)  = 1  + (−0.287 + 0.287i)3-s + (−0.877 − 0.479i)5-s + (−1.01 − 1.01i)7-s + 0.834i·9-s + 0.908i·11-s + (−0.871 − 0.871i)13-s + (0.389 − 0.114i)15-s + (0.0970 − 0.0970i)17-s + 0.952·19-s + 0.586·21-s + (1.03 − 1.03i)23-s + (0.539 + 0.841i)25-s + (−0.527 − 0.527i)27-s + 1.78i·29-s − 0.242i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.869 - 0.494i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9509385067\)
\(L(\frac12)\) \(\approx\) \(0.9509385067\)
\(L(\frac{7}{2})\) 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 \( 1 + (49.0 + 26.8i)T \)
good3 \( 1 + (4.47 - 4.47i)T - 243iT^{2} \)
7 \( 1 + (132. + 132. i)T + 1.68e4iT^{2} \)
11 \( 1 - 364. iT - 1.61e5T^{2} \)
13 \( 1 + (530. + 530. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-115. + 115. i)T - 1.41e6iT^{2} \)
19 \( 1 - 1.49e3T + 2.47e6T^{2} \)
23 \( 1 + (-2.63e3 + 2.63e3i)T - 6.43e6iT^{2} \)
29 \( 1 - 8.08e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.29e3iT - 2.86e7T^{2} \)
37 \( 1 + (5.68e3 - 5.68e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.02e4T + 1.15e8T^{2} \)
43 \( 1 + (-1.10e4 + 1.10e4i)T - 1.47e8iT^{2} \)
47 \( 1 + (-1.10e4 - 1.10e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (-2.31e4 - 2.31e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 6.31e3T + 7.14e8T^{2} \)
61 \( 1 + 3.30e4T + 8.44e8T^{2} \)
67 \( 1 + (-2.68e4 - 2.68e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 5.18e4iT - 1.80e9T^{2} \)
73 \( 1 + (5.52e3 + 5.52e3i)T + 2.07e9iT^{2} \)
79 \( 1 - 7.96e4T + 3.07e9T^{2} \)
83 \( 1 + (5.97e4 - 5.97e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.18e5iT - 5.58e9T^{2} \)
97 \( 1 + (-4.68e3 + 4.68e3i)T - 8.58e9iT^{2} \)
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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

−12.29911428104581571174138575178, −10.83934040270170638675239061887, −10.24058102694671396341689014381, −9.109636403475907176702323251785, −7.59634473451985459008438280650, −7.11445614004634945457955354269, −5.24670504371544021542228584861, −4.36673395116162975141082557347, −3.00982316574534078598139852208, −0.75772929975766275804108063018, 0.51720716537245668907318231384, 2.74562200000437408681752524810, 3.77222826844419424261900436877, 5.59835966650127069465731747054, 6.57636832764743872425567532841, 7.52258796240148594962108829751, 8.975317101741716491643918111958, 9.678725300948917786095922362824, 11.27244369269050159794036672868, 11.88621763953925627524352396348

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