Properties

Label 2-588-7.6-c2-0-5
Degree $2$
Conductor $588$
Sign $0.409 - 0.912i$
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  + 1.73i·3-s − 5.37i·5-s − 2.99·9-s − 8.59·11-s + 21.0i·13-s + 9.31·15-s + 5.48i·17-s + 7.24i·19-s + 28.0·23-s − 3.94·25-s − 5.19i·27-s + 40.3·29-s + 40.5i·31-s − 14.8i·33-s + 66.6·37-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.07i·5-s − 0.333·9-s − 0.781·11-s + 1.61i·13-s + 0.621·15-s + 0.322i·17-s + 0.381i·19-s + 1.21·23-s − 0.157·25-s − 0.192i·27-s + 1.39·29-s + 1.30i·31-s − 0.450i·33-s + 1.80·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.409 - 0.912i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ 0.409 - 0.912i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.512869610\)
\(L(\frac12)\) \(\approx\) \(1.512869610\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 + 5.37iT - 25T^{2} \)
11 \( 1 + 8.59T + 121T^{2} \)
13 \( 1 - 21.0iT - 169T^{2} \)
17 \( 1 - 5.48iT - 289T^{2} \)
19 \( 1 - 7.24iT - 361T^{2} \)
23 \( 1 - 28.0T + 529T^{2} \)
29 \( 1 - 40.3T + 841T^{2} \)
31 \( 1 - 40.5iT - 961T^{2} \)
37 \( 1 - 66.6T + 1.36e3T^{2} \)
41 \( 1 + 33.6iT - 1.68e3T^{2} \)
43 \( 1 - 0.932T + 1.84e3T^{2} \)
47 \( 1 - 85.6iT - 2.20e3T^{2} \)
53 \( 1 + 44.5T + 2.80e3T^{2} \)
59 \( 1 - 63.6iT - 3.48e3T^{2} \)
61 \( 1 + 32.0iT - 3.72e3T^{2} \)
67 \( 1 + 47.7T + 4.48e3T^{2} \)
71 \( 1 - 14.9T + 5.04e3T^{2} \)
73 \( 1 - 140. iT - 5.32e3T^{2} \)
79 \( 1 + 122.T + 6.24e3T^{2} \)
83 \( 1 + 33.1iT - 6.88e3T^{2} \)
89 \( 1 + 36.1iT - 7.92e3T^{2} \)
97 \( 1 + 16.2iT - 9.40e3T^{2} \)
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

−10.63230495318925358783170947646, −9.643465959650558812878768499046, −8.926091362794485802751711523596, −8.291465650574302823090911552099, −7.07720340918578777345128757168, −5.94461858689133057822367696766, −4.81951469666562413666924881204, −4.35310990858762260953602851673, −2.84667863718987127944023087408, −1.25967204220419888401513835563, 0.64013305115799304262993404935, 2.60170764961301906191198090889, 3.11472471768905324108377014330, 4.85427104560463411597489384600, 5.89285626785841312477036838337, 6.79205998293934506310967073838, 7.64831257020186594739170197219, 8.275275368137520670676619220688, 9.596441603681292004991477230453, 10.51396580494932769444838644652

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