↑

Properties

Label 2-504-504.331-c0-0-0
Degree $2$
Conductor $504$
Sign $0.400 + 0.916i$
Analytic cond. $0.251528$
Root an. cond. $0.501526$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s − i·5-s + 0.999i·6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)10-s − 11-s + (−0.499 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (−0.499 + 0.866i)14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s − i·5-s + 0.999i·6-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)10-s − 11-s + (−0.499 − 0.866i)12-s + (−0.866 + 0.5i)13-s + (−0.499 + 0.866i)14-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.400 + 0.916i$
Analytic conductor: \(0.251528\)
Root analytic conductor: \(0.501526\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :0),\ 0.400 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6981658840\)
\(L(\frac12)\) \(\approx\) \(0.6981658840\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−10.74197502192186914441838313559, −9.962834068592221164482482599679, −8.800236135261438244170639917646, −8.208563843639381803355439304649, −7.69434483208285858385423895009, −6.66474121562688648484078409956, −5.52461301882581725809246447208, −4.48828510979927263041744042250, −2.38015007545729846450083727022, −1.20648212692962212844108473145, 2.52439693746894059060497505507, 2.86590369608401469466776937956, 4.45627522563693860824705993114, 5.58497190282458153556853155659, 7.29132746648604362880211043838, 7.80007425149012691396194574755, 8.764880181615585234701256838756, 9.674193442231975396967771367032, 10.37014073222246349872694927199, 11.05323214785177871782226120736

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