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Properties

Label 2-160-20.19-c2-0-10
Degree $2$
Conductor $160$
Sign $0.891 + 0.453i$
Analytic cond. $4.35968$
Root an. cond. $2.08798$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.30·3-s + (−1.54 − 4.75i)5-s − 0.206·7-s + 19.1·9-s − 15.0i·11-s + 11.6i·13-s + (−8.20 − 25.2i)15-s + 18.1i·17-s + 19.3i·19-s − 1.09·21-s − 27.2·23-s + (−20.2 + 14.7i)25-s + 53.6·27-s + 44.4·29-s − 20.3i·31-s + ⋯
L(s)  = 1  + 1.76·3-s + (−0.309 − 0.950i)5-s − 0.0295·7-s + 2.12·9-s − 1.36i·11-s + 0.899i·13-s + (−0.547 − 1.68i)15-s + 1.07i·17-s + 1.02i·19-s − 0.0521·21-s − 1.18·23-s + (−0.808 + 0.588i)25-s + 1.98·27-s + 1.53·29-s − 0.657i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(4.35968\)
Root analytic conductor: \(2.08798\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :1),\ 0.891 + 0.453i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.30816 - 0.553449i\)
\(L(\frac12)\) \(\approx\) \(2.30816 - 0.553449i\)
\(L(2)\) 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 \( 1 + (1.54 + 4.75i)T \)
good3 \( 1 - 5.30T + 9T^{2} \)
7 \( 1 + 0.206T + 49T^{2} \)
11 \( 1 + 15.0iT - 121T^{2} \)
13 \( 1 - 11.6iT - 169T^{2} \)
17 \( 1 - 18.1iT - 289T^{2} \)
19 \( 1 - 19.3iT - 361T^{2} \)
23 \( 1 + 27.2T + 529T^{2} \)
29 \( 1 - 44.4T + 841T^{2} \)
31 \( 1 + 20.3iT - 961T^{2} \)
37 \( 1 - 18.1iT - 1.36e3T^{2} \)
41 \( 1 + 32.3T + 1.68e3T^{2} \)
43 \( 1 + 4.06T + 1.84e3T^{2} \)
47 \( 1 - 5.37T + 2.20e3T^{2} \)
53 \( 1 - 79.1iT - 2.80e3T^{2} \)
59 \( 1 + 83.3iT - 3.48e3T^{2} \)
61 \( 1 + 36.7T + 3.72e3T^{2} \)
67 \( 1 + 4.51T + 4.48e3T^{2} \)
71 \( 1 + 41.6iT - 5.04e3T^{2} \)
73 \( 1 + 41.5iT - 5.32e3T^{2} \)
79 \( 1 - 15.5iT - 6.24e3T^{2} \)
83 \( 1 + 50.9T + 6.88e3T^{2} \)
89 \( 1 - 10.8T + 7.92e3T^{2} \)
97 \( 1 - 12.1iT - 9.40e3T^{2} \)
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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

−12.80275514529515186570517747467, −11.81768200073188512061341471309, −10.24850653642115071047385240132, −9.207261515760116701404107925131, −8.342738836853369510506257245375, −7.972005323863507571666277638295, −6.20704691971912769390583455600, −4.33442150017082799852677176355, −3.40216736408634431585666688272, −1.68239090826553985607616724478, 2.32424707521927259411145723228, 3.23646027265340863072221257797, 4.59771461769173112010346235605, 6.85315100892267949315175415213, 7.55173810265370328568544335874, 8.510842370638382629424699196750, 9.732002430686469142716292506639, 10.30874329845858799139353273151, 11.86619443688838423281001098245, 12.96830543790264570502019252095

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