↑

Properties

Label 2-160-8.5-c5-0-19
Degree $2$
Conductor $160$
Sign $-0.965 - 0.258i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 29.2i·3-s − 25i·5-s + 168.·7-s − 610.·9-s − 514. i·11-s − 491. i·13-s − 730.·15-s + 183.·17-s − 1.25e3i·19-s − 4.91e3i·21-s + 423.·23-s − 625·25-s + 1.07e4i·27-s + 3.46e3i·29-s − 2.34e3·31-s + ⋯
L(s)  = 1  − 1.87i·3-s − 0.447i·5-s + 1.29·7-s − 2.51·9-s − 1.28i·11-s − 0.806i·13-s − 0.837·15-s + 0.153·17-s − 0.794i·19-s − 2.43i·21-s + 0.166·23-s − 0.200·25-s + 2.83i·27-s + 0.764i·29-s − 0.438·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.965 - 0.258i$
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: \((2,\ 160,\ (\ :5/2),\ -0.965 - 0.258i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.790884405\)
\(L(\frac12)\) \(\approx\) \(1.790884405\)
\(L(\frac{7}{2})\) 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 \( 1 + 29.2iT - 243T^{2} \)
7 \( 1 - 168.T + 1.68e4T^{2} \)
11 \( 1 + 514. iT - 1.61e5T^{2} \)
13 \( 1 + 491. iT - 3.71e5T^{2} \)
17 \( 1 - 183.T + 1.41e6T^{2} \)
19 \( 1 + 1.25e3iT - 2.47e6T^{2} \)
23 \( 1 - 423.T + 6.43e6T^{2} \)
29 \( 1 - 3.46e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.34e3T + 2.86e7T^{2} \)
37 \( 1 - 7.38e3iT - 6.93e7T^{2} \)
41 \( 1 - 4.24e3T + 1.15e8T^{2} \)
43 \( 1 - 1.51e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.53e4T + 2.29e8T^{2} \)
53 \( 1 + 1.13e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.19e4iT - 7.14e8T^{2} \)
61 \( 1 - 4.14e4iT - 8.44e8T^{2} \)
67 \( 1 + 6.65e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.62e4T + 1.80e9T^{2} \)
73 \( 1 + 8.62e4T + 2.07e9T^{2} \)
79 \( 1 - 1.97e4T + 3.07e9T^{2} \)
83 \( 1 - 8.37e3iT - 3.93e9T^{2} \)
89 \( 1 + 3.82e3T + 5.58e9T^{2} \)
97 \( 1 - 3.51e4T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.55079561937133758379008549100, −10.94444188980551307971537542502, −8.811641472457252223762955689491, −8.177653552581504540867704121020, −7.39085098624994449594889123833, −6.09292904594347810525471729016, −5.13522938910008304729951241053, −2.91408574495404865492455793270, −1.48833237632334566247811975479, −0.61052444913038110205597296122, 2.17256910376058636083587156765, 3.90519039956940846585024174254, 4.61998807302750808518361553242, 5.67836026410349321684152270450, 7.44960237006180340670075001892, 8.696227382254619662099106055415, 9.656673519556008430125619115370, 10.46797394600475440814214076479, 11.26416285918629161216537956640, 12.10778133283554885139389698133

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