↑

Properties

Label 2-99-11.3-c3-0-1
Degree $2$
Conductor $99$
Sign $-0.884 + 0.466i$
Analytic cond. $5.84118$
Root an. cond. $2.41685$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.61 + 2.62i)2-s + (3.70 − 11.4i)4-s + (4.69 + 3.40i)5-s + (−4.72 + 14.5i)7-s + (5.52 + 17.0i)8-s − 25.9·10-s + (7.07 + 35.7i)11-s + (−37.6 + 27.3i)13-s + (−21.1 − 65.0i)14-s + (12.9 + 9.40i)16-s + (−91.6 − 66.5i)17-s + (−34.2 − 105. i)19-s + (56.2 − 40.8i)20-s + (−119. − 110. i)22-s − 12.1·23-s + ⋯
L(s)  = 1  + (−1.27 + 0.929i)2-s + (0.463 − 1.42i)4-s + (0.419 + 0.304i)5-s + (−0.255 + 0.785i)7-s + (0.244 + 0.751i)8-s − 0.820·10-s + (0.194 + 0.980i)11-s + (−0.804 + 0.584i)13-s + (−0.403 − 1.24i)14-s + (0.202 + 0.146i)16-s + (−1.30 − 0.949i)17-s + (−0.413 − 1.27i)19-s + (0.629 − 0.457i)20-s + (−1.15 − 1.07i)22-s − 0.110·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.884 + 0.466i$
Analytic conductor: \(5.84118\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :3/2),\ -0.884 + 0.466i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0844997 - 0.341101i\)
\(L(\frac12)\) \(\approx\) \(0.0844997 - 0.341101i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.07 - 35.7i)T \)
good2 \( 1 + (3.61 - 2.62i)T + (2.47 - 7.60i)T^{2} \)
5 \( 1 + (-4.69 - 3.40i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (4.72 - 14.5i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (37.6 - 27.3i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (91.6 + 66.5i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (34.2 + 105. i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + 12.1T + 1.21e4T^{2} \)
29 \( 1 + (86.7 - 266. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (74.0 - 53.8i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (86.7 - 266. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (1.85 + 5.69i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + 126.T + 7.95e4T^{2} \)
47 \( 1 + (72.3 + 222. i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-7.78 + 5.65i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-96.3 + 296. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (-719. - 522. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + 589.T + 3.00e5T^{2} \)
71 \( 1 + (-212. - 154. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (52.0 - 160. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (403. - 293. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (-261. - 189. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 - 1.18e3T + 7.04e5T^{2} \)
97 \( 1 + (-938. + 681. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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

−14.46569602981937280913831630172, −13.01340141339736584814174089465, −11.73015033078859829204393207494, −10.34039652923989183932413373536, −9.338825793739287907168421075607, −8.786488393612752713343396208570, −7.11053862128640100635819867727, −6.63671999465136622552868569752, −4.98656730002981875769803147028, −2.24097078285833533721395480699, 0.28584594916189132593218092620, 1.97714137404701344110924073940, 3.76767045326732602545158903797, 5.91755753561279760710302660067, 7.62096144966296851485117458866, 8.626142435531437543544707598449, 9.688206033008023591943732745942, 10.50509442900965175839635078465, 11.35608081568487536903833395814, 12.62254198293903332881933434663

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