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Properties

Label 2-160-40.29-c5-0-9
Degree $2$
Conductor $160$
Sign $-0.144 - 0.989i$
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  + 16.0·3-s + (19.1 + 52.5i)5-s − 20.5i·7-s + 13.2·9-s + 619. i·11-s − 101.·13-s + (306. + 840. i)15-s − 527. i·17-s + 1.70e3i·19-s − 328. i·21-s + 1.54e3i·23-s + (−2.39e3 + 2.00e3i)25-s − 3.67e3·27-s − 4.30e3i·29-s − 201.·31-s + ⋯
L(s)  = 1  + 1.02·3-s + (0.342 + 0.939i)5-s − 0.158i·7-s + 0.0545·9-s + 1.54i·11-s − 0.166·13-s + (0.351 + 0.964i)15-s − 0.442i·17-s + 1.08i·19-s − 0.162i·21-s + 0.610i·23-s + (−0.766 + 0.642i)25-s − 0.970·27-s − 0.951i·29-s − 0.0376·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ -0.144 - 0.989i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.423348594\)
\(L(\frac12)\) \(\approx\) \(2.423348594\)
\(L(\frac{7}{2})\) 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 \( 1 + (-19.1 - 52.5i)T \)
good3 \( 1 - 16.0T + 243T^{2} \)
7 \( 1 + 20.5iT - 1.68e4T^{2} \)
11 \( 1 - 619. iT - 1.61e5T^{2} \)
13 \( 1 + 101.T + 3.71e5T^{2} \)
17 \( 1 + 527. iT - 1.41e6T^{2} \)
19 \( 1 - 1.70e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.54e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.30e3iT - 2.05e7T^{2} \)
31 \( 1 + 201.T + 2.86e7T^{2} \)
37 \( 1 - 1.24e4T + 6.93e7T^{2} \)
41 \( 1 + 1.39e4T + 1.15e8T^{2} \)
43 \( 1 - 1.62e4T + 1.47e8T^{2} \)
47 \( 1 - 2.99e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.35e4T + 4.18e8T^{2} \)
59 \( 1 + 1.37e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.03e4iT - 8.44e8T^{2} \)
67 \( 1 - 7.81e3T + 1.35e9T^{2} \)
71 \( 1 + 4.04e3T + 1.80e9T^{2} \)
73 \( 1 - 4.94e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.87e4T + 3.07e9T^{2} \)
83 \( 1 - 7.59e3T + 3.93e9T^{2} \)
89 \( 1 - 8.00e4T + 5.58e9T^{2} \)
97 \( 1 + 1.35e5iT - 8.58e9T^{2} \)
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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

−12.34094087394295700816015577866, −11.18311407393272318421327887206, −9.914911498676670261653902530397, −9.461378869006198914123582964007, −7.923591736907431870145530009645, −7.25921283854779614827838839184, −5.91925412457616769575833504592, −4.22363646454043466557528896658, −2.91237259980556644458736626890, −1.91453841767140208091847437569, 0.67505029982752725622319156806, 2.33341177716547615437117268722, 3.56676722598983946134134481566, 5.08006050770096952106185835907, 6.24509607469756255377758310713, 7.914916266948476948985921537049, 8.750504570173635888773004084699, 9.213026291958559974554873686511, 10.65348521063175379604700768103, 11.78301577338382774485275256876

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