↑

Properties

Label 2-66-11.9-c1-0-0
Degree $2$
Conductor $66$
Sign $0.0219 - 0.999i$
Analytic cond. $0.527012$
Root an. cond. $0.725956$
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  + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (1.11 + 3.44i)5-s + (−0.309 − 0.951i)6-s + (−0.5 − 0.363i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s − 3.61·10-s + (1.23 − 3.07i)11-s + 0.999·12-s + (1 − 3.07i)13-s + (0.5 − 0.363i)14-s + (−2.92 − 2.12i)15-s + (0.309 + 0.951i)16-s + (1.23 + 3.80i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.499 + 1.53i)5-s + (−0.126 − 0.388i)6-s + (−0.188 − 0.137i)7-s + (0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s − 1.14·10-s + (0.372 − 0.927i)11-s + 0.288·12-s + (0.277 − 0.853i)13-s + (0.133 − 0.0970i)14-s + (−0.755 − 0.549i)15-s + (0.0772 + 0.237i)16-s + (0.299 + 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66\)    =    \(2 \cdot 3 \cdot 11\)
Sign: $0.0219 - 0.999i$
Analytic conductor: \(0.527012\)
Root analytic conductor: \(0.725956\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{66} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 66,\ (\ :1/2),\ 0.0219 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531702 + 0.520139i\)
\(L(\frac12)\) \(\approx\) \(0.531702 + 0.520139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-1.23 + 3.07i)T \)
good5 \( 1 + (-1.11 - 3.44i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.5 + 0.363i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1 + 3.07i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.23 - 3.80i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.61 + 3.35i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 + (5.54 + 4.02i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.04 - 3.21i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.23 - 5.25i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.61 + 3.35i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 + (3.23 - 2.35i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.427 + 1.31i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.35 + 6.06i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.14 + 3.52i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 4.94T + 67T^{2} \)
71 \( 1 + (-1.85 - 5.70i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (11.7 + 8.55i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.11 - 6.51i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.5 - 1.53i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 + (2.97 - 9.14i)T + (-78.4 - 57.0i)T^{2} \)
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

−15.15884065177856563486526170386, −14.27473180399623685468540415574, −13.29787682734971404938038528191, −11.44025664610118655710534164403, −10.51637924901720960212812651175, −9.613273862287142170813815204638, −7.893444511474028259686950431622, −6.50644209949263145065873082498, −5.72883355505669511349423103519, −3.44652457399143454084560545978, 1.60160929682408434522196986924, 4.41395228321192014377078444129, 5.73535053188573036573677677582, 7.63301795319623868224634281725, 9.182679536335233007389406803725, 9.772810943573574442982284061122, 11.58486639682336151571018523220, 12.29642363825952095970689396523, 13.14508696841318593708384900156, 14.22136568048940733157363445960

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