↑

Properties

Label 2-160-20.19-c2-0-3
Degree $2$
Conductor $160$
Sign $0.0639 - 0.997i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.547·3-s + (−3.75 + 3.30i)5-s + 10.0·7-s − 8.69·9-s + 17.2i·11-s + 4.41i·13-s + (2.05 − 1.80i)15-s + 27.0i·17-s + 4.82i·19-s − 5.50·21-s + 15.2·23-s + (3.19 − 24.7i)25-s + 9.69·27-s − 2.38·29-s − 38.0i·31-s + ⋯
L(s)  = 1  − 0.182·3-s + (−0.750 + 0.660i)5-s + 1.43·7-s − 0.966·9-s + 1.56i·11-s + 0.339i·13-s + (0.137 − 0.120i)15-s + 1.58i·17-s + 0.254i·19-s − 0.262·21-s + 0.663·23-s + (0.127 − 0.991i)25-s + 0.359·27-s − 0.0821·29-s − 1.22i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0639 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0639 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.0639 - 0.997i$
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: \((2,\ 160,\ (\ :1),\ 0.0639 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.837567 + 0.785610i\)
\(L(\frac12)\) \(\approx\) \(0.837567 + 0.785610i\)
\(L(2)\) 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 \( 1 + (3.75 - 3.30i)T \)
good3 \( 1 + 0.547T + 9T^{2} \)
7 \( 1 - 10.0T + 49T^{2} \)
11 \( 1 - 17.2iT - 121T^{2} \)
13 \( 1 - 4.41iT - 169T^{2} \)
17 \( 1 - 27.0iT - 289T^{2} \)
19 \( 1 - 4.82iT - 361T^{2} \)
23 \( 1 - 15.2T + 529T^{2} \)
29 \( 1 + 2.38T + 841T^{2} \)
31 \( 1 + 38.0iT - 961T^{2} \)
37 \( 1 - 16.5iT - 1.36e3T^{2} \)
41 \( 1 + 13.3T + 1.68e3T^{2} \)
43 \( 1 + 59.7T + 1.84e3T^{2} \)
47 \( 1 - 62.4T + 2.20e3T^{2} \)
53 \( 1 + 71.5iT - 2.80e3T^{2} \)
59 \( 1 + 68.8iT - 3.48e3T^{2} \)
61 \( 1 - 40.9T + 3.72e3T^{2} \)
67 \( 1 - 51.0T + 4.48e3T^{2} \)
71 \( 1 - 40.4iT - 5.04e3T^{2} \)
73 \( 1 + 35.8iT - 5.32e3T^{2} \)
79 \( 1 - 126. iT - 6.24e3T^{2} \)
83 \( 1 - 75.1T + 6.88e3T^{2} \)
89 \( 1 + 106.T + 7.92e3T^{2} \)
97 \( 1 - 85.4iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.66941136901092075797314953917, −11.67451546531853879315693029322, −11.12382644742131197161189784992, −10.08154612170408255802929666739, −8.493632382136456761753237942305, −7.78603924896835348982168086216, −6.61614463761667962981384909145, −5.10498711924814821246612666551, −3.96011060333436230284728691096, −2.05973317917818800956403762456, 0.76803103504913028335978229597, 3.12188444324237871297378029229, 4.82129773229392065596726111265, 5.56448538343459781546639171263, 7.35768728863999549458113070669, 8.442497255702981498084116780031, 8.894615883087943026910877490147, 10.86657907474524888448657164217, 11.41520938165115077260959286125, 12.08350975502491877069673580238

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