Properties

Label 2-588-196.55-c1-0-33
Degree $2$
Conductor $588$
Sign $0.0745 + 0.997i$
Analytic cond. $4.69520$
Root an. cond. $2.16684$
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  + (−1.11 − 0.875i)2-s + (−0.900 − 0.433i)3-s + (0.465 + 1.94i)4-s + (0.698 − 1.45i)5-s + (0.620 + 1.27i)6-s + (1.90 − 1.84i)7-s + (1.18 − 2.56i)8-s + (0.623 + 0.781i)9-s + (−2.04 + 0.998i)10-s + (0.316 + 0.252i)11-s + (0.424 − 1.95i)12-s + (4.51 + 3.60i)13-s + (−3.72 + 0.378i)14-s + (−1.25 + 1.00i)15-s + (−3.56 + 1.81i)16-s + (0.218 + 0.0499i)17-s + ⋯
L(s)  = 1  + (−0.785 − 0.619i)2-s + (−0.520 − 0.250i)3-s + (0.232 + 0.972i)4-s + (0.312 − 0.648i)5-s + (0.253 + 0.518i)6-s + (0.718 − 0.695i)7-s + (0.419 − 0.907i)8-s + (0.207 + 0.260i)9-s + (−0.647 + 0.315i)10-s + (0.0955 + 0.0762i)11-s + (0.122 − 0.564i)12-s + (1.25 + 0.999i)13-s + (−0.994 + 0.101i)14-s + (−0.325 + 0.259i)15-s + (−0.891 + 0.452i)16-s + (0.0530 + 0.0121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0745 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0745 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.0745 + 0.997i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.0745 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.746082 - 0.692406i\)
\(L(\frac12)\) \(\approx\) \(0.746082 - 0.692406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 + 0.875i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (-1.90 + 1.84i)T \)
good5 \( 1 + (-0.698 + 1.45i)T + (-3.11 - 3.90i)T^{2} \)
11 \( 1 + (-0.316 - 0.252i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-4.51 - 3.60i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.218 - 0.0499i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 0.409T + 19T^{2} \)
23 \( 1 + (-0.361 + 0.0825i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.16 + 5.08i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + 1.31T + 31T^{2} \)
37 \( 1 + (-1.78 + 7.82i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-2.15 + 4.48i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (-3.03 - 6.30i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.19 + 1.49i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (2.01 + 8.82i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (4.95 - 2.38i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (6.33 + 1.44i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 + (-3.45 + 0.787i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (9.75 - 7.78i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 4.62iT - 79T^{2} \)
83 \( 1 + (0.804 + 1.00i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-2.38 + 1.90i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 - 11.4iT - 97T^{2} \)
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

−10.71043329492738839380185522263, −9.571397011978701819481237227341, −8.859081636225121188679435609091, −7.963698436941797325603662759934, −7.09412864384222224027266146150, −6.05512640530091582987328818265, −4.68045048224977303762983589081, −3.79777949034173931419665322118, −1.92659992601181852837334097476, −0.977699225954070745370150224614, 1.32019734738277648506749992950, 2.92997625741896107035958535367, 4.71503705587369225251116064861, 5.75546033331544783617702396925, 6.24220379794092300909887203736, 7.37046842461425170124152774933, 8.352620566540314756156847269452, 9.013652930505479560865066585716, 10.13470601270202843083032699746, 10.77912189207482120341106326391

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