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Properties

Label 2-160-40.29-c7-0-35
Degree $2$
Conductor $160$
Sign $-0.129 - 0.991i$
Analytic cond. $49.9816$
Root an. cond. $7.06976$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 46.1·3-s + (−271. + 68.1i)5-s − 1.08e3i·7-s − 52.7·9-s − 2.23e3i·11-s − 9.58e3·13-s + (1.25e4 − 3.14e3i)15-s − 2.58e4i·17-s − 1.92e4i·19-s + 5.02e4i·21-s − 9.05e4i·23-s + (6.88e4 − 3.69e4i)25-s + 1.03e5·27-s − 6.22e4i·29-s − 1.75e5·31-s + ⋯
L(s)  = 1  − 0.987·3-s + (−0.969 + 0.243i)5-s − 1.19i·7-s − 0.0241·9-s − 0.506i·11-s − 1.20·13-s + (0.958 − 0.240i)15-s − 1.27i·17-s − 0.645i·19-s + 1.18i·21-s − 1.55i·23-s + (0.881 − 0.472i)25-s + 1.01·27-s − 0.474i·29-s − 1.05·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.129 - 0.991i$
Analytic conductor: \(49.9816\)
Root analytic conductor: \(7.06976\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :7/2),\ -0.129 - 0.991i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.1707070843\)
\(L(\frac12)\) \(\approx\) \(0.1707070843\)
\(L(\frac{9}{2})\) 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 \( 1 + (271. - 68.1i)T \)
good3 \( 1 + 46.1T + 2.18e3T^{2} \)
7 \( 1 + 1.08e3iT - 8.23e5T^{2} \)
11 \( 1 + 2.23e3iT - 1.94e7T^{2} \)
13 \( 1 + 9.58e3T + 6.27e7T^{2} \)
17 \( 1 + 2.58e4iT - 4.10e8T^{2} \)
19 \( 1 + 1.92e4iT - 8.93e8T^{2} \)
23 \( 1 + 9.05e4iT - 3.40e9T^{2} \)
29 \( 1 + 6.22e4iT - 1.72e10T^{2} \)
31 \( 1 + 1.75e5T + 2.75e10T^{2} \)
37 \( 1 + 6.10e5T + 9.49e10T^{2} \)
41 \( 1 + 2.10e5T + 1.94e11T^{2} \)
43 \( 1 - 1.78e5T + 2.71e11T^{2} \)
47 \( 1 + 8.54e4iT - 5.06e11T^{2} \)
53 \( 1 + 2.29e5T + 1.17e12T^{2} \)
59 \( 1 - 9.46e5iT - 2.48e12T^{2} \)
61 \( 1 - 1.68e6iT - 3.14e12T^{2} \)
67 \( 1 - 2.17e6T + 6.06e12T^{2} \)
71 \( 1 + 2.01e6T + 9.09e12T^{2} \)
73 \( 1 - 2.01e6iT - 1.10e13T^{2} \)
79 \( 1 - 1.35e6T + 1.92e13T^{2} \)
83 \( 1 + 9.39e6T + 2.71e13T^{2} \)
89 \( 1 - 5.56e6T + 4.42e13T^{2} \)
97 \( 1 + 9.80e6iT - 8.07e13T^{2} \)
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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

−10.95077123771986577961812326457, −10.20851919099346559488278727095, −8.684928247953880170068545841979, −7.33292566878992651322228175433, −6.81091513722821066110770773882, −5.21108959491655111123936123832, −4.29727943768168616707800820170, −2.87916307702809529738469495756, −0.56667929072593769222327223962, −0.098301455233866284864609583685, 1.78638728131336636449863613141, 3.48789103256259026366173913450, 4.98874131736470077692645329822, 5.67875647803669523964895452447, 7.03490295917155399706588111709, 8.163747366295455604604780088390, 9.219913741978357185888005246078, 10.50338216914669739925415524569, 11.54397122508689192331502075202, 12.23669771416876854114147431103

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