↑

Properties

Label 2-99-11.9-c3-0-6
Degree $2$
Conductor $99$
Sign $0.890 - 0.454i$
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  + (−1.58 + 4.87i)2-s + (−14.7 − 10.7i)4-s + (−4.61 − 14.2i)5-s + (9.98 + 7.25i)7-s + (42.5 − 30.9i)8-s + 76.5·10-s + (31.7 + 18.0i)11-s + (9.43 − 29.0i)13-s + (−51.1 + 37.1i)14-s + (38.1 + 117. i)16-s + (−40.8 − 125. i)17-s + (18.0 − 13.1i)19-s + (−84.2 + 259. i)20-s + (−138. + 126. i)22-s + 158.·23-s + ⋯
L(s)  = 1  + (−0.559 + 1.72i)2-s + (−1.84 − 1.34i)4-s + (−0.412 − 1.27i)5-s + (0.539 + 0.391i)7-s + (1.88 − 1.36i)8-s + 2.42·10-s + (0.869 + 0.494i)11-s + (0.201 − 0.619i)13-s + (−0.976 + 0.709i)14-s + (0.595 + 1.83i)16-s + (−0.582 − 1.79i)17-s + (0.217 − 0.158i)19-s + (−0.942 + 2.90i)20-s + (−1.33 + 1.22i)22-s + 1.43·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.890 - 0.454i$
Analytic conductor: \(5.84118\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :3/2),\ 0.890 - 0.454i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.922721 + 0.221666i\)
\(L(\frac12)\) \(\approx\) \(0.922721 + 0.221666i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-31.7 - 18.0i)T \)
good2 \( 1 + (1.58 - 4.87i)T + (-6.47 - 4.70i)T^{2} \)
5 \( 1 + (4.61 + 14.2i)T + (-101. + 73.4i)T^{2} \)
7 \( 1 + (-9.98 - 7.25i)T + (105. + 326. i)T^{2} \)
13 \( 1 + (-9.43 + 29.0i)T + (-1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (40.8 + 125. i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (-18.0 + 13.1i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 - 158.T + 1.21e4T^{2} \)
29 \( 1 + (38.6 + 28.0i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (-21.0 + 64.7i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (128. + 93.1i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (-231. + 168. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 - 103.T + 7.95e4T^{2} \)
47 \( 1 + (375. - 272. i)T + (3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (-1.86 + 5.75i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (179. + 130. i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (84.7 + 260. i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 - 187.T + 3.00e5T^{2} \)
71 \( 1 + (-141. - 435. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-218. - 158. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (-152. + 469. i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (-83.7 - 257. i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 - 77.4T + 7.04e5T^{2} \)
97 \( 1 + (392. - 1.20e3i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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

−13.82920707120826123824373184663, −12.69162005370951295279336374489, −11.40185925879906821844504497511, −9.418562288397030544328856190513, −8.948947639252177219662674100063, −7.905182409725002839885121548570, −6.88828490474412573861260519119, −5.33605051221466289895648084319, −4.62698964880483738061866174343, −0.75211968789748615925303563157, 1.51919654341815645418098350620, 3.20600194945583447738890877035, 4.18313597820001289577512323753, 6.69176960219810338791169208926, 8.177192566783864332384209870349, 9.224307965135393223023619932551, 10.63411183314017917506110329128, 10.99299455251314374490313432976, 11.81358826187868313018674596222, 13.04746720620630081425650332964

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