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Properties

Label 2-160-8.3-c4-0-6
Degree $2$
Conductor $160$
Sign $0.457 + 0.889i$
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  − 13.1·3-s − 11.1i·5-s + 61.6i·7-s + 91.5·9-s − 182.·11-s + 141. i·13-s + 146. i·15-s + 40.3·17-s + 321.·19-s − 809. i·21-s − 816. i·23-s − 125.·25-s − 138.·27-s − 1.19e3i·29-s − 361. i·31-s + ⋯
L(s)  = 1  − 1.45·3-s − 0.447i·5-s + 1.25i·7-s + 1.12·9-s − 1.50·11-s + 0.838i·13-s + 0.652i·15-s + 0.139·17-s + 0.890·19-s − 1.83i·21-s − 1.54i·23-s − 0.200·25-s − 0.189·27-s − 1.42i·29-s − 0.376i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.457 + 0.889i$
Analytic conductor: \(16.5391\)
Root analytic conductor: \(4.06684\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :2),\ 0.457 + 0.889i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.541155 - 0.330128i\)
\(L(\frac12)\) \(\approx\) \(0.541155 - 0.330128i\)
\(L(3)\) 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 \( 1 + 11.1iT \)
good3 \( 1 + 13.1T + 81T^{2} \)
7 \( 1 - 61.6iT - 2.40e3T^{2} \)
11 \( 1 + 182.T + 1.46e4T^{2} \)
13 \( 1 - 141. iT - 2.85e4T^{2} \)
17 \( 1 - 40.3T + 8.35e4T^{2} \)
19 \( 1 - 321.T + 1.30e5T^{2} \)
23 \( 1 + 816. iT - 2.79e5T^{2} \)
29 \( 1 + 1.19e3iT - 7.07e5T^{2} \)
31 \( 1 + 361. iT - 9.23e5T^{2} \)
37 \( 1 - 0.0209iT - 1.87e6T^{2} \)
41 \( 1 + 872.T + 2.82e6T^{2} \)
43 \( 1 - 3.12e3T + 3.41e6T^{2} \)
47 \( 1 + 2.65e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.95e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.41e3T + 1.21e7T^{2} \)
61 \( 1 + 2.25e3iT - 1.38e7T^{2} \)
67 \( 1 - 2.35e3T + 2.01e7T^{2} \)
71 \( 1 - 660. iT - 2.54e7T^{2} \)
73 \( 1 - 5.94e3T + 2.83e7T^{2} \)
79 \( 1 + 7.62e3iT - 3.89e7T^{2} \)
83 \( 1 - 3.52e3T + 4.74e7T^{2} \)
89 \( 1 - 5.94e3T + 6.27e7T^{2} \)
97 \( 1 - 679.T + 8.85e7T^{2} \)
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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

−12.07672979004041737588129054602, −11.19843954172903644838279398603, −10.19225713283822550846498858367, −9.024600930255266165135208329798, −7.79693810143605969583491180185, −6.29157815842528301609014980867, −5.49611788690343499880098653431, −4.65811724952130210213876130767, −2.38927414778842235297033757381, −0.38998675022357131634049251470, 0.935077174325273511037742052353, 3.29672324930304273493700856051, 4.96318125525992865330498844009, 5.73666691958518896022383652299, 7.11655664347921083336644614386, 7.76919808293364890403250845857, 9.813348804580365998433956354305, 10.71372145770503831804412093343, 11.01711840154364694617738529203, 12.31675456902400426398017459960

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