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Properties

Label 2-160-8.5-c5-0-11
Degree $2$
Conductor $160$
Sign $0.892 - 0.451i$
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  + 11.5i·3-s + 25i·5-s + 231.·7-s + 108.·9-s − 559. i·11-s − 107. i·13-s − 289.·15-s − 441.·17-s − 1.87e3i·19-s + 2.68e3i·21-s + 3.83e3·23-s − 625·25-s + 4.07e3i·27-s − 3.36e3i·29-s + 7.95e3·31-s + ⋯
L(s)  = 1  + 0.743i·3-s + 0.447i·5-s + 1.78·7-s + 0.446·9-s − 1.39i·11-s − 0.177i·13-s − 0.332·15-s − 0.370·17-s − 1.19i·19-s + 1.32i·21-s + 1.51·23-s − 0.200·25-s + 1.07i·27-s − 0.744i·29-s + 1.48·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.892 - 0.451i$
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: \((2,\ 160,\ (\ :5/2),\ 0.892 - 0.451i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.584912669\)
\(L(\frac12)\) \(\approx\) \(2.584912669\)
\(L(\frac{7}{2})\) 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 \( 1 - 11.5iT - 243T^{2} \)
7 \( 1 - 231.T + 1.68e4T^{2} \)
11 \( 1 + 559. iT - 1.61e5T^{2} \)
13 \( 1 + 107. iT - 3.71e5T^{2} \)
17 \( 1 + 441.T + 1.41e6T^{2} \)
19 \( 1 + 1.87e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.83e3T + 6.43e6T^{2} \)
29 \( 1 + 3.36e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.95e3T + 2.86e7T^{2} \)
37 \( 1 - 1.06e4iT - 6.93e7T^{2} \)
41 \( 1 + 9.96e3T + 1.15e8T^{2} \)
43 \( 1 - 925. iT - 1.47e8T^{2} \)
47 \( 1 + 8.06e3T + 2.29e8T^{2} \)
53 \( 1 - 7.95e3iT - 4.18e8T^{2} \)
59 \( 1 - 1.68e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.12e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.36e4iT - 1.35e9T^{2} \)
71 \( 1 - 8.86e3T + 1.80e9T^{2} \)
73 \( 1 - 5.55e4T + 2.07e9T^{2} \)
79 \( 1 + 6.94e4T + 3.07e9T^{2} \)
83 \( 1 + 1.02e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.24e4T + 5.58e9T^{2} \)
97 \( 1 + 8.86e4T + 8.58e9T^{2} \)
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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

−11.53499916785083291231992598353, −11.14989378313178184543247051434, −10.24795392150860514690446833176, −8.891463815758988122555720027463, −8.079026495306775781758177769916, −6.77819704593542297631131526270, −5.22036575709607839084603854137, −4.41248458104132802673559011345, −2.89059668548709839337734682333, −1.10797962504319227587912313179, 1.25858451898299969435584111929, 1.99137701497580300320493208889, 4.35297402792269357794478057325, 5.14256678497861971909139737875, 6.85404214395803894294248591670, 7.70366063577363843359288323215, 8.564102880697476237197370550109, 9.899744484265712022952310810817, 11.06601476097986074597632366932, 12.10742810782193494051699691854

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