↑

Properties

Label 2-99-3.2-c2-0-7
Degree $2$
Conductor $99$
Sign $-0.816 - 0.577i$
Analytic cond. $2.69755$
Root an. cond. $1.64242$
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  − 3.89i·2-s − 11.2·4-s − 6.10i·5-s + 2.61·7-s + 28.0i·8-s − 23.7·10-s + 3.31i·11-s − 7.69·13-s − 10.1i·14-s + 64.7·16-s − 27.5i·17-s + 3.63·19-s + 68.3i·20-s + 12.9·22-s − 22.4i·23-s + ⋯
L(s)  = 1  − 1.94i·2-s − 2.80·4-s − 1.22i·5-s + 0.372·7-s + 3.51i·8-s − 2.37·10-s + 0.301i·11-s − 0.592·13-s − 0.727i·14-s + 4.04·16-s − 1.62i·17-s + 0.191·19-s + 3.41i·20-s + 0.587·22-s − 0.975i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ -0.816 - 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.300183 + 0.944456i\)
\(L(\frac12)\) \(\approx\) \(0.300183 + 0.944456i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 - 3.31iT \)
good2 \( 1 + 3.89iT - 4T^{2} \)
5 \( 1 + 6.10iT - 25T^{2} \)
7 \( 1 - 2.61T + 49T^{2} \)
13 \( 1 + 7.69T + 169T^{2} \)
17 \( 1 + 27.5iT - 289T^{2} \)
19 \( 1 - 3.63T + 361T^{2} \)
23 \( 1 + 22.4iT - 529T^{2} \)
29 \( 1 + 16.9iT - 841T^{2} \)
31 \( 1 - 3.26T + 961T^{2} \)
37 \( 1 - 15.0T + 1.36e3T^{2} \)
41 \( 1 - 40.2iT - 1.68e3T^{2} \)
43 \( 1 - 69.4T + 1.84e3T^{2} \)
47 \( 1 - 21.7iT - 2.20e3T^{2} \)
53 \( 1 - 12.1iT - 2.80e3T^{2} \)
59 \( 1 - 34.0iT - 3.48e3T^{2} \)
61 \( 1 - 61.3T + 3.72e3T^{2} \)
67 \( 1 - 54.6T + 4.48e3T^{2} \)
71 \( 1 + 11.5iT - 5.04e3T^{2} \)
73 \( 1 - 41.7T + 5.32e3T^{2} \)
79 \( 1 - 96.9T + 6.24e3T^{2} \)
83 \( 1 + 89.8iT - 6.88e3T^{2} \)
89 \( 1 - 117. iT - 7.92e3T^{2} \)
97 \( 1 + 7.47T + 9.40e3T^{2} \)
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

−12.65902652952014235677837701384, −12.01753005438616860633037261460, −11.10673191536934246316801719668, −9.782938421686321812765596479412, −9.143482818695892755906070309803, −7.997074413207911447498856343528, −5.11511974884193308201814717549, −4.40982244585631747860046535376, −2.53155448642799038955089705164, −0.823160333623156396346803876693, 3.79492696364512226621247730019, 5.40649851662024829122327852253, 6.47718829890368437453285585247, 7.41041881104449343495509255648, 8.342770510673619173089392046098, 9.649217507312782310149006067291, 10.82100118184408126476245537429, 12.63543057073519561837569746809, 13.82752486767713642483861722517, 14.59351767275600789002739965400

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