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Properties

Label 2-160-8.5-c5-0-13
Degree $2$
Conductor $160$
Sign $-0.0379 + 0.999i$
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  − 18.7i·3-s + 25i·5-s + 107.·7-s − 109.·9-s − 272. i·11-s − 198. i·13-s + 469.·15-s + 2.06e3·17-s + 1.89e3i·19-s − 2.02e3i·21-s + 987.·23-s − 625·25-s − 2.49e3i·27-s − 8.01e3i·29-s − 827.·31-s + ⋯
L(s)  = 1  − 1.20i·3-s + 0.447i·5-s + 0.829·7-s − 0.452·9-s − 0.678i·11-s − 0.325i·13-s + 0.538·15-s + 1.73·17-s + 1.20i·19-s − 0.999i·21-s + 0.389·23-s − 0.200·25-s − 0.659i·27-s − 1.76i·29-s − 0.154·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0379 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0379 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.0379 + 0.999i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ -0.0379 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.168548789\)
\(L(\frac12)\) \(\approx\) \(2.168548789\)
\(L(\frac{7}{2})\) 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 \( 1 - 25iT \)
good3 \( 1 + 18.7iT - 243T^{2} \)
7 \( 1 - 107.T + 1.68e4T^{2} \)
11 \( 1 + 272. iT - 1.61e5T^{2} \)
13 \( 1 + 198. iT - 3.71e5T^{2} \)
17 \( 1 - 2.06e3T + 1.41e6T^{2} \)
19 \( 1 - 1.89e3iT - 2.47e6T^{2} \)
23 \( 1 - 987.T + 6.43e6T^{2} \)
29 \( 1 + 8.01e3iT - 2.05e7T^{2} \)
31 \( 1 + 827.T + 2.86e7T^{2} \)
37 \( 1 + 9.42e3iT - 6.93e7T^{2} \)
41 \( 1 + 8.22e3T + 1.15e8T^{2} \)
43 \( 1 + 9.30e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.38e4T + 2.29e8T^{2} \)
53 \( 1 - 2.77e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.51e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.64e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.85e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.10e4T + 1.80e9T^{2} \)
73 \( 1 - 1.86e4T + 2.07e9T^{2} \)
79 \( 1 - 7.55e4T + 3.07e9T^{2} \)
83 \( 1 - 1.25e5iT - 3.93e9T^{2} \)
89 \( 1 - 3.03e4T + 5.58e9T^{2} \)
97 \( 1 - 1.56e4T + 8.58e9T^{2} \)
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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

−11.90994968198313637948918134300, −10.89421894015479374536041833194, −9.770646272679073228854407939107, −8.008172441713448161038963641878, −7.81814537580731995966336514959, −6.42280467502948310775648461789, −5.43359722920422425353354023690, −3.55815803645921964863397319429, −2.00184520601613364558815717172, −0.812333429012405914020304403537, 1.38772710423946496734483801737, 3.33409452741340317732084688745, 4.71841184765186675352112704200, 5.18180160510757385868484675776, 7.04544023614676914730362864852, 8.308278003008287067442571447065, 9.345076426365460052177829730855, 10.12873203381046433562453432118, 11.12812282800311555164704427073, 12.08422184116632829463425231928

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