↑

Properties

Label 2-588-84.11-c1-0-10
Degree $2$
Conductor $588$
Sign $-0.964 - 0.264i$
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  + (1.04 + 0.951i)2-s + (−1.73 + 0.0590i)3-s + (0.189 + 1.99i)4-s + (0.301 − 0.174i)5-s + (−1.86 − 1.58i)6-s + (−1.69 + 2.26i)8-s + (2.99 − 0.204i)9-s + (0.481 + 0.104i)10-s + (−1.95 + 3.38i)11-s + (−0.445 − 3.43i)12-s − 2.93·13-s + (−0.512 + 0.319i)15-s + (−3.92 + 0.755i)16-s + (3.38 + 1.95i)17-s + (3.32 + 2.63i)18-s + (−4.83 + 2.78i)19-s + ⋯
L(s)  = 1  + (0.739 + 0.672i)2-s + (−0.999 + 0.0340i)3-s + (0.0948 + 0.995i)4-s + (0.135 − 0.0779i)5-s + (−0.762 − 0.647i)6-s + (−0.599 + 0.800i)8-s + (0.997 − 0.0681i)9-s + (0.152 + 0.0331i)10-s + (−0.588 + 1.01i)11-s + (−0.128 − 0.991i)12-s − 0.815·13-s + (−0.132 + 0.0825i)15-s + (−0.982 + 0.188i)16-s + (0.819 + 0.473i)17-s + (0.783 + 0.620i)18-s + (−1.10 + 0.639i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.964 - 0.264i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.964 - 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146049 + 1.08432i\)
\(L(\frac12)\) \(\approx\) \(0.146049 + 1.08432i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 - 0.951i)T \)
3 \( 1 + (1.73 - 0.0590i)T \)
7 \( 1 \)
good5 \( 1 + (-0.301 + 0.174i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.95 - 3.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 + (-3.38 - 1.95i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.83 - 2.78i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.09 - 1.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.75iT - 29T^{2} \)
31 \( 1 + (2.28 + 1.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.319 + 0.553i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.57iT - 41T^{2} \)
43 \( 1 - 2.51iT - 43T^{2} \)
47 \( 1 + (-2.18 - 3.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.49 + 0.860i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.12 + 7.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.04 - 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.553 + 0.319i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + (3.93 - 6.82i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.46 + 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.94T + 83T^{2} \)
89 \( 1 + (-9.12 + 5.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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

−11.34903279581903978617702998097, −10.21173105866376837469646674862, −9.539407859841829698479265895664, −7.972287775747095605777381932279, −7.43245439044092852345445382717, −6.37703770112929569498891027676, −5.60333008030909644976982571501, −4.77495727015981766015527317966, −3.88781419597427856702372463006, −2.15298543533025153778441835904, 0.52192533918759812859140788914, 2.25399232739511220465246223538, 3.56495268934452796628261685285, 4.86371252734562955997952557642, 5.42614732590063780940765374561, 6.40767983959140263596705851334, 7.25943326645111292950550976909, 8.729504073140334876292407879759, 9.859846198199241667891581614635, 10.64632179517927611222776770789

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