Properties

Label 2-600-3.2-c2-0-33
Degree $2$
Conductor $600$
Sign $-0.942 + 0.333i$
Analytic cond. $16.3488$
Root an. cond. $4.04336$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 2.82i)3-s + 6·7-s + (−7.00 + 5.65i)9-s − 5.65i·11-s − 10·13-s − 22.6i·17-s + 2·19-s + (−6 − 16.9i)21-s − 11.3i·23-s + (23.0 + 14.1i)27-s − 16.9i·29-s − 22·31-s + (−16.0 + 5.65i)33-s + 6·37-s + (10 + 28.2i)39-s + ⋯
L(s)  = 1  + (−0.333 − 0.942i)3-s + 0.857·7-s + (−0.777 + 0.628i)9-s − 0.514i·11-s − 0.769·13-s − 1.33i·17-s + 0.105·19-s + (−0.285 − 0.808i)21-s − 0.491i·23-s + (0.851 + 0.523i)27-s − 0.585i·29-s − 0.709·31-s + (−0.484 + 0.171i)33-s + 0.162·37-s + (0.256 + 0.725i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.942 + 0.333i$
Analytic conductor: \(16.3488\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1),\ -0.942 + 0.333i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9819634912\)
\(L(\frac12)\) \(\approx\) \(0.9819634912\)
\(L(2)\) 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 \( 1 + (1 + 2.82i)T \)
5 \( 1 \)
good7 \( 1 - 6T + 49T^{2} \)
11 \( 1 + 5.65iT - 121T^{2} \)
13 \( 1 + 10T + 169T^{2} \)
17 \( 1 + 22.6iT - 289T^{2} \)
19 \( 1 - 2T + 361T^{2} \)
23 \( 1 + 11.3iT - 529T^{2} \)
29 \( 1 + 16.9iT - 841T^{2} \)
31 \( 1 + 22T + 961T^{2} \)
37 \( 1 - 6T + 1.36e3T^{2} \)
41 \( 1 + 33.9iT - 1.68e3T^{2} \)
43 \( 1 + 82T + 1.84e3T^{2} \)
47 \( 1 - 67.8iT - 2.20e3T^{2} \)
53 \( 1 + 62.2iT - 2.80e3T^{2} \)
59 \( 1 + 73.5iT - 3.48e3T^{2} \)
61 \( 1 + 86T + 3.72e3T^{2} \)
67 \( 1 + 2T + 4.48e3T^{2} \)
71 \( 1 - 124. iT - 5.04e3T^{2} \)
73 \( 1 + 82T + 5.32e3T^{2} \)
79 \( 1 - 10T + 6.24e3T^{2} \)
83 \( 1 + 73.5iT - 6.88e3T^{2} \)
89 \( 1 - 33.9iT - 7.92e3T^{2} \)
97 \( 1 - 94T + 9.40e3T^{2} \)
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

−10.17044126896561567826761969354, −9.053177211766155184430251379533, −8.105317707445382906380271735112, −7.43469559228312728443862544050, −6.55865421523715370040808558026, −5.43593877596170304469492259870, −4.70791545687891635746465022033, −2.96985268159926244867399890483, −1.81352770494242502634152029841, −0.37341211478045611868561431226, 1.73702581797980401787472406948, 3.32831464461184648419747785828, 4.45643443198741955659301751784, 5.14112255413689336688448336473, 6.14612600969401866276264379837, 7.36125129788992719779666357132, 8.333118716657602660312684393582, 9.194970347199334460215534819667, 10.12829414983079771637782043312, 10.70244785868761495917228250825

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