↑

Properties

Label 2-588-4.3-c2-0-32
Degree $2$
Conductor $588$
Sign $0.247 + 0.968i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.249 − 1.98i)2-s − 1.73i·3-s + (−3.87 + 0.989i)4-s − 4.12·5-s + (−3.43 + 0.431i)6-s + (2.92 + 7.44i)8-s − 2.99·9-s + (1.02 + 8.17i)10-s + 15.1i·11-s + (1.71 + 6.71i)12-s + 5.62·13-s + 7.13i·15-s + (14.0 − 7.66i)16-s + 29.0·17-s + (0.747 + 5.95i)18-s − 7.26i·19-s + ⋯
L(s)  = 1  + (−0.124 − 0.992i)2-s − 0.577i·3-s + (−0.968 + 0.247i)4-s − 0.824·5-s + (−0.572 + 0.0719i)6-s + (0.366 + 0.930i)8-s − 0.333·9-s + (0.102 + 0.817i)10-s + 1.37i·11-s + (0.142 + 0.559i)12-s + 0.432·13-s + 0.475i·15-s + (0.877 − 0.479i)16-s + 1.71·17-s + (0.0415 + 0.330i)18-s − 0.382i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.247 + 0.968i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 0.247 + 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.218354890\)
\(L(\frac12)\) \(\approx\) \(1.218354890\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.249 + 1.98i)T \)
3 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 + 4.12T + 25T^{2} \)
11 \( 1 - 15.1iT - 121T^{2} \)
13 \( 1 - 5.62T + 169T^{2} \)
17 \( 1 - 29.0T + 289T^{2} \)
19 \( 1 + 7.26iT - 361T^{2} \)
23 \( 1 + 25.3iT - 529T^{2} \)
29 \( 1 + 8.59T + 841T^{2} \)
31 \( 1 - 16.6iT - 961T^{2} \)
37 \( 1 + 22.4T + 1.36e3T^{2} \)
41 \( 1 - 49.2T + 1.68e3T^{2} \)
43 \( 1 - 70.5iT - 1.84e3T^{2} \)
47 \( 1 + 50.1iT - 2.20e3T^{2} \)
53 \( 1 - 100.T + 2.80e3T^{2} \)
59 \( 1 + 76.7iT - 3.48e3T^{2} \)
61 \( 1 - 111.T + 3.72e3T^{2} \)
67 \( 1 - 89.7iT - 4.48e3T^{2} \)
71 \( 1 + 52.9iT - 5.04e3T^{2} \)
73 \( 1 - 32.8T + 5.32e3T^{2} \)
79 \( 1 - 16.2iT - 6.24e3T^{2} \)
83 \( 1 + 25.6iT - 6.88e3T^{2} \)
89 \( 1 - 96.7T + 7.92e3T^{2} \)
97 \( 1 + 54.1T + 9.40e3T^{2} \)
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

−10.34024575959447863394641942294, −9.623320265491963322605800541993, −8.503198285632464246496385645657, −7.80526110144031629521885070348, −6.98038745351483869923721500490, −5.48441038355121936193877752425, −4.39875646003661238822689785614, −3.42537399924405413807764758666, −2.17568004902516932937927334080, −0.840145579518705191095586421371, 0.76992053487380635539541989036, 3.47224546873493154432994801366, 3.94979957505780923745055290114, 5.48407313353043420087863306921, 5.84958523891232855347871247230, 7.31164123987412413542625996759, 8.006734466117354765508753768033, 8.717484742594249290142560668120, 9.640695286960387567965996712808, 10.52196896395272574832222826945

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