↑

Properties

Label 2-588-84.23-c1-0-62
Degree $2$
Conductor $588$
Sign $0.523 + 0.852i$
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  + (1.34 − 0.443i)2-s + (1.25 − 1.19i)3-s + (1.60 − 1.19i)4-s + (1.08 + 0.626i)5-s + (1.15 − 2.15i)6-s + (1.63 − 2.31i)8-s + (0.148 − 2.99i)9-s + (1.73 + 0.360i)10-s + (2.52 + 4.38i)11-s + (0.594 − 3.41i)12-s − 4.41·13-s + (2.11 − 0.509i)15-s + (1.16 − 3.82i)16-s + (−5.06 + 2.92i)17-s + (−1.12 − 4.08i)18-s + (1.32 + 0.765i)19-s + ⋯
L(s)  = 1  + (0.949 − 0.313i)2-s + (0.724 − 0.689i)3-s + (0.803 − 0.595i)4-s + (0.485 + 0.280i)5-s + (0.471 − 0.881i)6-s + (0.576 − 0.817i)8-s + (0.0494 − 0.998i)9-s + (0.548 + 0.114i)10-s + (0.762 + 1.32i)11-s + (0.171 − 0.985i)12-s − 1.22·13-s + (0.544 − 0.131i)15-s + (0.291 − 0.956i)16-s + (−1.22 + 0.709i)17-s + (−0.266 − 0.963i)18-s + (0.304 + 0.175i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.523 + 0.852i$
Analytic conductor: \(4.69520\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1/2),\ 0.523 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.96080 - 1.65689i\)
\(L(\frac12)\) \(\approx\) \(2.96080 - 1.65689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 0.443i)T \)
3 \( 1 + (-1.25 + 1.19i)T \)
7 \( 1 \)
good5 \( 1 + (-1.08 - 0.626i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.52 - 4.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.41T + 13T^{2} \)
17 \( 1 + (5.06 - 2.92i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.32 - 0.765i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.156 + 0.271i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.53iT - 29T^{2} \)
31 \( 1 + (4.24 - 2.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.66 + 4.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.97iT - 41T^{2} \)
43 \( 1 - 3.18iT - 43T^{2} \)
47 \( 1 + (-5.74 + 9.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.0 - 6.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.98 - 6.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.151 - 0.262i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.13 + 5.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.53T + 71T^{2} \)
73 \( 1 + (-0.707 - 1.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6 + 3.46i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.78T + 83T^{2} \)
89 \( 1 + (4.15 + 2.39i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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

−10.56152344070184405632091773724, −9.743075591320994229362450917187, −8.971782983717432139679204441126, −7.46805147260200555113301112909, −6.93816656251355263606906144388, −6.11131668146201304213173345030, −4.77964427000816139212229580345, −3.80001068434317190801279043124, −2.44147398428270346328305133570, −1.78415682012435393407604323615, 2.19675553109084682265939121049, 3.20840512114471264820710029670, 4.30001779050805400553538854268, 5.13071170481104892577859429816, 6.09717589184676968161537360376, 7.21658685532637840230536431915, 8.165742286806709238410601543770, 9.131470177830242631299506028302, 9.775835390423300595891409040685, 11.13878792692581192551731104733

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