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Properties

Label 2-99-11.5-c3-0-4
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

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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: $\chi_{99} (82, \cdot )$
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 \( 1 + (392. + 1.20e3i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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