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Properties

Label 2-99-11.3-c3-0-4
Degree $2$
Conductor $99$
Sign $0.621 - 0.783i$
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  + (1.39 − 1.01i)2-s + (−1.54 + 4.76i)4-s + (3.78 + 2.75i)5-s + (−2.91 + 8.97i)7-s + (6.95 + 21.3i)8-s + 8.09·10-s + (20.8 + 29.9i)11-s + (−4.36 + 3.16i)13-s + (5.04 + 15.5i)14-s + (−0.939 − 0.682i)16-s + (68.6 + 49.8i)17-s + (−15.7 − 48.4i)19-s + (−18.9 + 13.7i)20-s + (59.6 + 20.5i)22-s + 52.9·23-s + ⋯
L(s)  = 1  + (0.494 − 0.359i)2-s + (−0.193 + 0.595i)4-s + (0.338 + 0.246i)5-s + (−0.157 + 0.484i)7-s + (0.307 + 0.945i)8-s + 0.256·10-s + (0.572 + 0.819i)11-s + (−0.0930 + 0.0676i)13-s + (0.0962 + 0.296i)14-s + (−0.0146 − 0.0106i)16-s + (0.979 + 0.711i)17-s + (−0.189 − 0.584i)19-s + (−0.212 + 0.154i)20-s + (0.577 + 0.199i)22-s + 0.480·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.621 - 0.783i$
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: \((2,\ 99,\ (\ :3/2),\ 0.621 - 0.783i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.74802 + 0.844803i\)
\(L(\frac12)\) \(\approx\) \(1.74802 + 0.844803i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-20.8 - 29.9i)T \)
good2 \( 1 + (-1.39 + 1.01i)T + (2.47 - 7.60i)T^{2} \)
5 \( 1 + (-3.78 - 2.75i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (2.91 - 8.97i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (4.36 - 3.16i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-68.6 - 49.8i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (15.7 + 48.4i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 - 52.9T + 1.21e4T^{2} \)
29 \( 1 + (-35.2 + 108. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (67.7 - 49.2i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (47.0 - 144. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (56.9 + 175. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + 46.4T + 7.95e4T^{2} \)
47 \( 1 + (120. + 370. i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-462. + 336. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (24.4 - 75.2i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (584. + 424. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 - 608.T + 3.00e5T^{2} \)
71 \( 1 + (-740. - 538. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (-186. + 573. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (-298. + 216. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (-797. - 579. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 - 1.15e3T + 7.04e5T^{2} \)
97 \( 1 + (478. - 347. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−13.45028158035892843837904566938, −12.41702276478503258718022608606, −11.81907516287808221030057547472, −10.43489062497329870943098369671, −9.234975674565767722484226911202, −8.053650146343555596284019139384, −6.66525153679442087417661534095, −5.14325897101030735619610388746, −3.74787565407092752980227896322, −2.25502104823458872780668015815, 1.06963482785208819969545338532, 3.63739156235709203020585204622, 5.14285591681447946331404397706, 6.13359944450733873357708996610, 7.38807581628054247742677887034, 9.030866078188901265432637944387, 9.963521991545711654355383055948, 11.06367878001109718675575423498, 12.46926645368263375543629797042, 13.55900447301162316614800623418

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