↑

Properties

Label 2-579-579.50-c0-0-0
Degree $2$
Conductor $579$
Sign $0.599 + 0.800i$
Analytic cond. $0.288958$
Root an. cond. $0.537548$
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  + (0.707 − 0.707i)3-s + (−0.707 + 0.707i)4-s − 1.84i·7-s − 1.00i·9-s + 1.00i·12-s + (0.0761 + 0.382i)13-s − 1.00i·16-s + (0.923 + 0.617i)19-s + (−1.30 − 1.30i)21-s + (0.382 + 0.923i)25-s + (−0.707 − 0.707i)27-s + (1.30 + 1.30i)28-s + (0.541 + 1.30i)31-s + (0.707 + 0.707i)36-s + (−1.92 + 0.382i)37-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.707 + 0.707i)4-s − 1.84i·7-s − 1.00i·9-s + 1.00i·12-s + (0.0761 + 0.382i)13-s − 1.00i·16-s + (0.923 + 0.617i)19-s + (−1.30 − 1.30i)21-s + (0.382 + 0.923i)25-s + (−0.707 − 0.707i)27-s + (1.30 + 1.30i)28-s + (0.541 + 1.30i)31-s + (0.707 + 0.707i)36-s + (−1.92 + 0.382i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $0.599 + 0.800i$
Analytic conductor: \(0.288958\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 579,\ (\ :0),\ 0.599 + 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9717901188\)
\(L(\frac12)\) \(\approx\) \(0.9717901188\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 + 0.707i)T \)
193 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (-0.382 - 0.923i)T^{2} \)
7 \( 1 + 1.84iT - T^{2} \)
11 \( 1 + (0.382 - 0.923i)T^{2} \)
13 \( 1 + (-0.0761 - 0.382i)T + (-0.923 + 0.382i)T^{2} \)
17 \( 1 + (-0.382 + 0.923i)T^{2} \)
19 \( 1 + (-0.923 - 0.617i)T + (0.382 + 0.923i)T^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.923 - 0.382i)T^{2} \)
31 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \)
41 \( 1 + (0.382 - 0.923i)T^{2} \)
43 \( 1 + 0.765iT - T^{2} \)
47 \( 1 + (0.923 - 0.382i)T^{2} \)
53 \( 1 + (0.923 - 0.382i)T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (1.38 - 0.923i)T + (0.382 - 0.923i)T^{2} \)
67 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (-0.382 - 0.923i)T^{2} \)
73 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (0.923 + 0.382i)T^{2} \)
97 \( 1 + (-0.707 - 0.707i)T^{2} \)
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

−10.68967237278062222234349299725, −9.797443121221412493992539608883, −8.918208916377262470736536258159, −8.041100940365388673867877916355, −7.31162732325766808046014562357, −6.76771546332949007178175604903, −5.01171687689733111043503803020, −3.82718189272894313645341078114, −3.27941957663915329388288790567, −1.29135306380624535174567383913, 2.15809695894677694770600879533, 3.26081094306817718324266763834, 4.69855014432002677666067823392, 5.34090552842342896339922910889, 6.22626889222095033888532717858, 7.907237249879931011983286402857, 8.727029074557566516741181534242, 9.244658188858360184414935659211, 9.927058916932733148147056234811, 10.86254396475716581285268560049

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