↑

Properties

Label 2-320-20.19-c2-0-15
Degree $2$
Conductor $320$
Sign $0.660 + 0.750i$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.660 + 0.750i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ 0.660 + 0.750i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.92266 - 0.869464i\)
\(L(\frac12)\) \(\approx\) \(1.92266 - 0.869464i\)
\(L(2)\) 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

−11.10077498469117137108595971087, −10.65602088763955023952895311188, −9.066076205159032752321309939840, −8.539401497487640372432599371614, −7.83026505717034957678128699984, −5.93502536525789068510806994772, −5.55939668885473855558186139960, −4.17836906548806416964698102786, −2.56426444219594265530665803970, −1.09580178642028382534424643042, 1.77490469078102151063089691467, 2.82701785799769273841867787575, 4.66382556665392459024294867711, 5.42302245103450914137966992103, 6.86748835311173454217908647757, 7.60071492345207739685682153824, 8.821500293604171034357438115293, 9.637204370048977218538586631747, 10.69429816219289110287336658960, 11.46864765387487182325294613528

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