↑

Properties

Label 2-147-3.2-c2-0-9
Degree $2$
Conductor $147$
Sign $0.607 - 0.794i$
Analytic cond. $4.00545$
Root an. cond. $2.00136$
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  + 1.30i·2-s + (1.82 − 2.38i)3-s + 2.29·4-s + 7.37i·5-s + (3.11 + 2.38i)6-s + 8.22i·8-s + (−2.35 − 8.68i)9-s − 9.64·10-s + 2.61i·11-s + (4.17 − 5.45i)12-s + 6.35·13-s + (17.5 + 13.4i)15-s − 1.58·16-s − 12.1i·17-s + (11.3 − 3.07i)18-s + 10.2·19-s + ⋯
L(s)  = 1  + 0.653i·2-s + (0.607 − 0.794i)3-s + 0.572·4-s + 1.47i·5-s + (0.519 + 0.397i)6-s + 1.02i·8-s + (−0.261 − 0.965i)9-s − 0.964·10-s + 0.237i·11-s + (0.348 − 0.454i)12-s + 0.488·13-s + (1.17 + 0.896i)15-s − 0.0989·16-s − 0.714i·17-s + (0.630 − 0.170i)18-s + 0.538·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.607 - 0.794i$
Analytic conductor: \(4.00545\)
Root analytic conductor: \(2.00136\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1),\ 0.607 - 0.794i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.79435 + 0.886476i\)
\(L(\frac12)\) \(\approx\) \(1.79435 + 0.886476i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.82 + 2.38i)T \)
7 \( 1 \)
good2 \( 1 - 1.30iT - 4T^{2} \)
5 \( 1 - 7.37iT - 25T^{2} \)
11 \( 1 - 2.61iT - 121T^{2} \)
13 \( 1 - 6.35T + 169T^{2} \)
17 \( 1 + 12.1iT - 289T^{2} \)
19 \( 1 - 10.2T + 361T^{2} \)
23 \( 1 + 4.30iT - 529T^{2} \)
29 \( 1 + 17.3iT - 841T^{2} \)
31 \( 1 + 39.2T + 961T^{2} \)
37 \( 1 - 41.0T + 1.36e3T^{2} \)
41 \( 1 + 30.2iT - 1.68e3T^{2} \)
43 \( 1 + 55.8T + 1.84e3T^{2} \)
47 \( 1 + 39.9iT - 2.20e3T^{2} \)
53 \( 1 + 105. iT - 2.80e3T^{2} \)
59 \( 1 - 41.3iT - 3.48e3T^{2} \)
61 \( 1 - 20.4T + 3.72e3T^{2} \)
67 \( 1 + 27.1T + 4.48e3T^{2} \)
71 \( 1 - 67.8iT - 5.04e3T^{2} \)
73 \( 1 + 60.7T + 5.32e3T^{2} \)
79 \( 1 + 63.2T + 6.24e3T^{2} \)
83 \( 1 - 89.9iT - 6.88e3T^{2} \)
89 \( 1 - 63.1iT - 7.92e3T^{2} \)
97 \( 1 + 19.1T + 9.40e3T^{2} \)
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

−13.25073918791277581677068665566, −11.80865054860429064207273240732, −11.14822986127373993373874235515, −9.874138348300289869759896577188, −8.404542243257939181111331994855, −7.30487689185896192622579463284, −6.86728597733091940298506732235, −5.79682748128285366490312629479, −3.33050601032802348676233267243, −2.20905898155817180146579231469, 1.52611910406020479279001794858, 3.29552981207969484399096606044, 4.49340662316691190554468125770, 5.83010773448385164116885535990, 7.70763154756520582793872119703, 8.780014771629126925233383166147, 9.576973479460728984602637293336, 10.65922559320751000293314384034, 11.57671555209282639755668175200, 12.70765378998503819466748281996

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