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Properties

Label 2-160-8.5-c7-0-14
Degree $2$
Conductor $160$
Sign $0.850 - 0.525i$
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  + 47.3i·3-s + 125i·5-s − 608.·7-s − 52.1·9-s − 6.83e3i·11-s + 5.23e3i·13-s − 5.91e3·15-s + 2.66e4·17-s − 5.21e4i·19-s − 2.88e4i·21-s + 6.62e4·23-s − 1.56e4·25-s + 1.01e5i·27-s − 2.83e3i·29-s − 8.27e4·31-s + ⋯
L(s)  = 1  + 1.01i·3-s + 0.447i·5-s − 0.671·7-s − 0.0238·9-s − 1.54i·11-s + 0.661i·13-s − 0.452·15-s + 1.31·17-s − 1.74i·19-s − 0.679i·21-s + 1.13·23-s − 0.199·25-s + 0.987i·27-s − 0.0216i·29-s − 0.498·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(49.9816\)
Root analytic conductor: \(7.06976\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :7/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.994818445\)
\(L(\frac12)\) \(\approx\) \(1.994818445\)
\(L(\frac{9}{2})\) 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 \( 1 - 125iT \)
good3 \( 1 - 47.3iT - 2.18e3T^{2} \)
7 \( 1 + 608.T + 8.23e5T^{2} \)
11 \( 1 + 6.83e3iT - 1.94e7T^{2} \)
13 \( 1 - 5.23e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.66e4T + 4.10e8T^{2} \)
19 \( 1 + 5.21e4iT - 8.93e8T^{2} \)
23 \( 1 - 6.62e4T + 3.40e9T^{2} \)
29 \( 1 + 2.83e3iT - 1.72e10T^{2} \)
31 \( 1 + 8.27e4T + 2.75e10T^{2} \)
37 \( 1 + 3.92e5iT - 9.49e10T^{2} \)
41 \( 1 - 5.22e5T + 1.94e11T^{2} \)
43 \( 1 - 3.07e5iT - 2.71e11T^{2} \)
47 \( 1 + 2.82e5T + 5.06e11T^{2} \)
53 \( 1 - 1.23e6iT - 1.17e12T^{2} \)
59 \( 1 - 1.44e6iT - 2.48e12T^{2} \)
61 \( 1 + 6.34e5iT - 3.14e12T^{2} \)
67 \( 1 + 1.13e6iT - 6.06e12T^{2} \)
71 \( 1 - 1.83e6T + 9.09e12T^{2} \)
73 \( 1 - 2.75e6T + 1.10e13T^{2} \)
79 \( 1 - 8.21e6T + 1.92e13T^{2} \)
83 \( 1 - 3.77e6iT - 2.71e13T^{2} \)
89 \( 1 - 4.96e6T + 4.42e13T^{2} \)
97 \( 1 - 4.13e6T + 8.07e13T^{2} \)
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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

−11.23131687370762215770198210533, −10.78475225793641176914379306478, −9.534536425790165133082772599634, −8.982436899216372609760270094903, −7.41149734188391221464565946459, −6.25450674620527708346515032340, −5.06801431964690880817055776730, −3.71522068465300724367350956666, −2.89480811131420586725433991282, −0.73148873966556176330440897718, 0.902164096291673327364466759564, 1.92380437003263366910884224153, 3.50735857736657456962975122981, 5.05499270080623404425180843983, 6.29664139951447736279740711164, 7.37153954707201272306003339473, 8.061965254892346884924137191293, 9.613189379023310064004026196848, 10.22884099253241919978127426605, 11.94138123259174985771860537760

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