↑

Properties

Label 2-504-168.11-c1-0-25
Degree $2$
Conductor $504$
Sign $0.784 + 0.620i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 + 0.0663i)2-s + (1.99 + 0.187i)4-s + (−1.02 − 1.78i)5-s + (−1.24 − 2.33i)7-s + (2.80 + 0.396i)8-s + (−1.33 − 2.58i)10-s + (5.24 + 3.03i)11-s − 1.77i·13-s + (−1.60 − 3.37i)14-s + (3.92 + 0.746i)16-s + (0.786 + 0.453i)17-s + (−3.37 − 5.84i)19-s + (−1.71 − 3.73i)20-s + (7.21 + 4.62i)22-s + (−0.351 − 0.608i)23-s + ⋯
L(s)  = 1  + (0.998 + 0.0469i)2-s + (0.995 + 0.0937i)4-s + (−0.459 − 0.796i)5-s + (−0.471 − 0.882i)7-s + (0.990 + 0.140i)8-s + (−0.421 − 0.816i)10-s + (1.58 + 0.913i)11-s − 0.493i·13-s + (−0.429 − 0.903i)14-s + (0.982 + 0.186i)16-s + (0.190 + 0.110i)17-s + (−0.774 − 1.34i)19-s + (−0.383 − 0.835i)20-s + (1.53 + 0.987i)22-s + (−0.0732 − 0.126i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.784 + 0.620i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.784 + 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38261 - 0.827840i\)
\(L(\frac12)\) \(\approx\) \(2.38261 - 0.827840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0663i)T \)
3 \( 1 \)
7 \( 1 + (1.24 + 2.33i)T \)
good5 \( 1 + (1.02 + 1.78i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.24 - 3.03i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.77iT - 13T^{2} \)
17 \( 1 + (-0.786 - 0.453i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.37 + 5.84i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.351 + 0.608i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.52T + 29T^{2} \)
31 \( 1 + (-7.31 - 4.22i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.07 - 3.50i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.6iT - 41T^{2} \)
43 \( 1 + 4.69T + 43T^{2} \)
47 \( 1 + (-4.42 - 7.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.31 - 5.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.11 + 0.645i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.95 - 4.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.562 - 0.974i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.43T + 71T^{2} \)
73 \( 1 + (-2.08 + 3.61i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.6 + 6.71i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.71iT - 83T^{2} \)
89 \( 1 + (4.08 - 2.35i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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

−11.02591046431144169972088152708, −10.08535024853974852282306281953, −9.050095929857465836136106742169, −7.915063770161002899208415944651, −6.90395191193058569070244808190, −6.31809874333254816836046144151, −4.68423320444011018503612088218, −4.34543892439281802253609005936, −3.13582844608023042532002595291, −1.30550487301068789818677494058, 1.99847176569766203977554949043, 3.42589412140633548369129747257, 3.91836597887823600896898671128, 5.54823616113901905208992685748, 6.33257575154818710570561987735, 6.96745532969382158286342456336, 8.238706786163774792960713867314, 9.268514117203617774163459944186, 10.40157426327788659630591017726, 11.32560213659231072960151334278

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