Properties

Label 2-160-40.13-c4-0-2
Degree $2$
Conductor $160$
Sign $-0.997 + 0.0714i$
Analytic cond. $16.5391$
Root an. cond. $4.06684$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.10 + 3.10i)3-s + (−18.7 + 16.4i)5-s + (52.9 − 52.9i)7-s + 61.7i·9-s + 95.9i·11-s + (23.7 − 23.7i)13-s + (7.09 − 109. i)15-s + (−187. + 187. i)17-s − 312.·19-s + 328. i·21-s + (−658. − 658. i)23-s + (80.7 − 619. i)25-s + (−442. − 442. i)27-s − 1.30e3·29-s + 215.·31-s + ⋯
L(s)  = 1  + (−0.344 + 0.344i)3-s + (−0.751 + 0.659i)5-s + (1.08 − 1.08i)7-s + 0.762i·9-s + 0.793i·11-s + (0.140 − 0.140i)13-s + (0.0315 − 0.486i)15-s + (−0.650 + 0.650i)17-s − 0.866·19-s + 0.745i·21-s + (−1.24 − 1.24i)23-s + (0.129 − 0.991i)25-s + (−0.607 − 0.607i)27-s − 1.54·29-s + 0.223·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0714i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.997 + 0.0714i$
Analytic conductor: \(16.5391\)
Root analytic conductor: \(4.06684\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :2),\ -0.997 + 0.0714i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0131461 - 0.367598i\)
\(L(\frac12)\) \(\approx\) \(0.0131461 - 0.367598i\)
\(L(3)\) 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 \( 1 + (18.7 - 16.4i)T \)
good3 \( 1 + (3.10 - 3.10i)T - 81iT^{2} \)
7 \( 1 + (-52.9 + 52.9i)T - 2.40e3iT^{2} \)
11 \( 1 - 95.9iT - 1.46e4T^{2} \)
13 \( 1 + (-23.7 + 23.7i)T - 2.85e4iT^{2} \)
17 \( 1 + (187. - 187. i)T - 8.35e4iT^{2} \)
19 \( 1 + 312.T + 1.30e5T^{2} \)
23 \( 1 + (658. + 658. i)T + 2.79e5iT^{2} \)
29 \( 1 + 1.30e3T + 7.07e5T^{2} \)
31 \( 1 - 215.T + 9.23e5T^{2} \)
37 \( 1 + (-886. - 886. i)T + 1.87e6iT^{2} \)
41 \( 1 + 1.20e3T + 2.82e6T^{2} \)
43 \( 1 + (413. - 413. i)T - 3.41e6iT^{2} \)
47 \( 1 + (738. - 738. i)T - 4.87e6iT^{2} \)
53 \( 1 + (2.78e3 - 2.78e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 1.38e3T + 1.21e7T^{2} \)
61 \( 1 + 2.79e3iT - 1.38e7T^{2} \)
67 \( 1 + (-436. - 436. i)T + 2.01e7iT^{2} \)
71 \( 1 + 1.91e3T + 2.54e7T^{2} \)
73 \( 1 + (-5.03e3 - 5.03e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 1.41e3iT - 3.89e7T^{2} \)
83 \( 1 + (-7.06e3 + 7.06e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.28e4iT - 6.27e7T^{2} \)
97 \( 1 + (1.95e3 - 1.95e3i)T - 8.85e7iT^{2} \)
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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

−12.63042145683166494872505451009, −11.34701761542630030941025765042, −10.80900297978791543848126356588, −10.11312064729014947715120435181, −8.239818164447016266729894111989, −7.64003825954334631913374231406, −6.43168151398788309392129231829, −4.66945975915913059658270915290, −4.07563992855886742540715447071, −2.00681956537818217769260317559, 0.14306980843626456607977262211, 1.78072657009129798082203328334, 3.76167673560275020680638883487, 5.14040779915177819536122347067, 6.12726070238033291160523830797, 7.63312996513368692791112012322, 8.572233705337020070651699937249, 9.322620049165842616384174172761, 11.28023214811361426130009518399, 11.58218855044885689246911953027

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