↑

Properties

Label 2-588-28.27-c1-0-10
Degree $2$
Conductor $588$
Sign $-0.889 - 0.456i$
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  + (0.639 + 1.26i)2-s + 3-s + (−1.18 + 1.61i)4-s + 3.10i·5-s + (0.639 + 1.26i)6-s + (−2.79 − 0.457i)8-s + 9-s + (−3.91 + 1.98i)10-s + 5.34i·11-s + (−1.18 + 1.61i)12-s − 3.92i·13-s + 3.10i·15-s + (−1.20 − 3.81i)16-s − 5.68i·17-s + (0.639 + 1.26i)18-s + 0.170·19-s + ⋯
L(s)  = 1  + (0.452 + 0.891i)2-s + 0.577·3-s + (−0.590 + 0.806i)4-s + 1.38i·5-s + (0.261 + 0.514i)6-s + (−0.986 − 0.161i)8-s + 0.333·9-s + (−1.23 + 0.628i)10-s + 1.61i·11-s + (−0.340 + 0.465i)12-s − 1.08i·13-s + 0.801i·15-s + (−0.302 − 0.953i)16-s − 1.37i·17-s + (0.150 + 0.297i)18-s + 0.0391·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.889 - 0.456i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ -0.889 - 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.454424 + 1.87892i\)
\(L(\frac12)\) \(\approx\) \(0.454424 + 1.87892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.639 - 1.26i)T \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 - 3.10iT - 5T^{2} \)
11 \( 1 - 5.34iT - 11T^{2} \)
13 \( 1 + 3.92iT - 13T^{2} \)
17 \( 1 + 5.68iT - 17T^{2} \)
19 \( 1 - 0.170T + 19T^{2} \)
23 \( 1 - 6.13iT - 23T^{2} \)
29 \( 1 + 1.96T + 29T^{2} \)
31 \( 1 + 1.53T + 31T^{2} \)
37 \( 1 - 8.93T + 37T^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 - 2.38iT - 43T^{2} \)
47 \( 1 - 2.29T + 47T^{2} \)
53 \( 1 + 1.19T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 6.58iT - 61T^{2} \)
67 \( 1 + 9.64iT - 67T^{2} \)
71 \( 1 - 2.04iT - 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 - 14.3iT - 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 5.81iT - 89T^{2} \)
97 \( 1 + 2.32iT - 97T^{2} \)
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

−11.08577504654003959542494174223, −9.834248448298831343304024128675, −9.473570113447444353083442077763, −7.973050071475679424202122128044, −7.37610650352963955077030897025, −6.86088307241409991113457894937, −5.65277757225727405926306917259, −4.56202324418564071408411569413, −3.35610605715024885322895864928, −2.54417186647354953063277071120, 0.926183751250272192020558030601, 2.20528143873944627441766394952, 3.68662746564805681775306864138, 4.36755446180014705832123709125, 5.49384348273927912630392646595, 6.38838274303896857315292377486, 8.199825405194030919207891737261, 8.741541413193852640976534200098, 9.292760585778762930772931925037, 10.42751302866121898129240293547

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