↑

Properties

Label 2-264-88.21-c2-0-5
Degree $2$
Conductor $264$
Sign $-0.987 + 0.157i$
Analytic cond. $7.19347$
Root an. cond. $2.68206$
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  + (0.613 + 1.90i)2-s − 1.73i·3-s + (−3.24 + 2.33i)4-s + 6.76i·5-s + (3.29 − 1.06i)6-s + 3.01i·7-s + (−6.44 − 4.74i)8-s − 2.99·9-s + (−12.8 + 4.15i)10-s + (−7.83 − 7.72i)11-s + (4.04 + 5.62i)12-s + 0.226·13-s + (−5.73 + 1.84i)14-s + 11.7·15-s + (5.07 − 15.1i)16-s + 22.7i·17-s + ⋯
L(s)  = 1  + (0.306 + 0.951i)2-s − 0.577i·3-s + (−0.811 + 0.584i)4-s + 1.35i·5-s + (0.549 − 0.177i)6-s + 0.430i·7-s + (−0.805 − 0.592i)8-s − 0.333·9-s + (−1.28 + 0.415i)10-s + (−0.712 − 0.702i)11-s + (0.337 + 0.468i)12-s + 0.0174·13-s + (−0.409 + 0.132i)14-s + 0.781·15-s + (0.317 − 0.948i)16-s + 1.33i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $-0.987 + 0.157i$
Analytic conductor: \(7.19347\)
Root analytic conductor: \(2.68206\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 264,\ (\ :1),\ -0.987 + 0.157i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0759748 - 0.961779i\)
\(L(\frac12)\) \(\approx\) \(0.0759748 - 0.961779i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.613 - 1.90i)T \)
3 \( 1 + 1.73iT \)
11 \( 1 + (7.83 + 7.72i)T \)
good5 \( 1 - 6.76iT - 25T^{2} \)
7 \( 1 - 3.01iT - 49T^{2} \)
13 \( 1 - 0.226T + 169T^{2} \)
17 \( 1 - 22.7iT - 289T^{2} \)
19 \( 1 + 35.6T + 361T^{2} \)
23 \( 1 - 31.6T + 529T^{2} \)
29 \( 1 + 26.0T + 841T^{2} \)
31 \( 1 + 53.8T + 961T^{2} \)
37 \( 1 - 40.5iT - 1.36e3T^{2} \)
41 \( 1 - 34.4iT - 1.68e3T^{2} \)
43 \( 1 - 36.6T + 1.84e3T^{2} \)
47 \( 1 - 67.4T + 2.20e3T^{2} \)
53 \( 1 + 18.8iT - 2.80e3T^{2} \)
59 \( 1 + 17.3iT - 3.48e3T^{2} \)
61 \( 1 + 4.73T + 3.72e3T^{2} \)
67 \( 1 - 90.4iT - 4.48e3T^{2} \)
71 \( 1 - 81.0T + 5.04e3T^{2} \)
73 \( 1 + 3.38iT - 5.32e3T^{2} \)
79 \( 1 - 118. iT - 6.24e3T^{2} \)
83 \( 1 - 31.9T + 6.88e3T^{2} \)
89 \( 1 - 33.6T + 7.92e3T^{2} \)
97 \( 1 + 142.T + 9.40e3T^{2} \)
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

−12.69662732722515594615489101089, −11.18162785043161680962996576645, −10.54427395604258360744651160484, −8.968963107356084751422428784684, −8.147042553844950837268182366418, −7.13700054724852393268195528016, −6.34003488298536627412631534494, −5.54112825874967173341100735420, −3.79451775884275628158034409864, −2.56859178336595081403751399821, 0.43065764365438104997272262936, 2.23111721240028842064608995332, 3.94032320713814127772838990449, 4.79288479083290935955946575851, 5.52510154513550756859068466003, 7.44376019160150423136350343706, 9.024628383578031710552178926080, 9.142748050946564171717214887424, 10.51296016124481660093315459614, 11.06942772910502516698633327712

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