↑

Properties

Label 2-504-63.25-c1-0-23
Degree $2$
Conductor $504$
Sign $-0.174 + 0.984i$
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  + (1.22 − 1.22i)3-s + (1.76 − 3.05i)5-s + (−2.63 + 0.176i)7-s + (0.0136 − 2.99i)9-s + (1.16 + 2.00i)11-s + (−2.35 − 4.08i)13-s + (−1.56 − 5.90i)15-s + (−0.636 + 1.10i)17-s + (2.78 + 4.82i)19-s + (−3.02 + 3.44i)21-s + (1.64 − 2.85i)23-s + (−3.72 − 6.45i)25-s + (−3.64 − 3.69i)27-s + (−4.32 + 7.48i)29-s + 8.51·31-s + ⋯
L(s)  = 1  + (0.708 − 0.705i)3-s + (0.789 − 1.36i)5-s + (−0.997 + 0.0666i)7-s + (0.00456 − 0.999i)9-s + (0.349 + 0.605i)11-s + (−0.654 − 1.13i)13-s + (−0.405 − 1.52i)15-s + (−0.154 + 0.267i)17-s + (0.638 + 1.10i)19-s + (−0.660 + 0.751i)21-s + (0.343 − 0.595i)23-s + (−0.745 − 1.29i)25-s + (−0.702 − 0.711i)27-s + (−0.802 + 1.38i)29-s + 1.52·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.174 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14485 - 1.36606i\)
\(L(\frac12)\) \(\approx\) \(1.14485 - 1.36606i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.22 + 1.22i)T \)
7 \( 1 + (2.63 - 0.176i)T \)
good5 \( 1 + (-1.76 + 3.05i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.16 - 2.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.35 + 4.08i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.636 - 1.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.78 - 4.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.64 + 2.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.32 - 7.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.51T + 31T^{2} \)
37 \( 1 + (2.84 + 4.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.66 - 2.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0444 + 0.0769i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.05T + 47T^{2} \)
53 \( 1 + (-3.41 + 5.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.99T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 - 6.12T + 67T^{2} \)
71 \( 1 - 1.30T + 71T^{2} \)
73 \( 1 + (-6.64 + 11.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 + (5.90 - 10.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.561 - 0.972i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.50 - 6.07i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.26951611512110811582158581679, −9.619657463601392068372644543925, −8.945402436099835904364572956218, −8.096573916531653451483465360674, −7.06903349111291232235834273052, −6.03190573074529352949366528918, −5.12382806838497243424755858430, −3.67054275537132533821577914141, −2.37324284658495912956885063337, −1.03242871853349178344881301191, 2.41748346004629257790055127358, 3.08069485773589193084378686384, 4.22750376851851341525382489922, 5.66753623317022779859416683684, 6.70903266442377330787420276612, 7.32017074951787772369692816432, 8.815948729077811000631636811764, 9.661069592820360688751539575511, 9.936802121206894566726428204798, 11.03835773104849985627242131323

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