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Properties

Label 2-99-33.32-c3-0-11
Degree $2$
Conductor $99$
Sign $0.174 + 0.984i$
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  + 3·2-s + 4-s − 19.7i·5-s − 16.9i·7-s − 21·8-s − 59.3i·10-s + (33 + 15.5i)11-s + 29.6i·13-s − 50.9i·14-s − 71·16-s + 126·17-s − 89.0i·19-s − 19.7i·20-s + (99 + 46.6i)22-s + 120. i·23-s + ⋯
L(s)  = 1  + 1.06·2-s + 0.125·4-s − 1.77i·5-s − 0.916i·7-s − 0.928·8-s − 1.87i·10-s + (0.904 + 0.426i)11-s + 0.633i·13-s − 0.971i·14-s − 1.10·16-s + 1.79·17-s − 1.07i·19-s − 0.221i·20-s + (0.959 + 0.452i)22-s + 1.08i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.174 + 0.984i$
Analytic conductor: \(5.84118\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :3/2),\ 0.174 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.75729 - 1.47389i\)
\(L(\frac12)\) \(\approx\) \(1.75729 - 1.47389i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-33 - 15.5i)T \)
good2 \( 1 - 3T + 8T^{2} \)
5 \( 1 + 19.7iT - 125T^{2} \)
7 \( 1 + 16.9iT - 343T^{2} \)
13 \( 1 - 29.6iT - 2.19e3T^{2} \)
17 \( 1 - 126T + 4.91e3T^{2} \)
19 \( 1 + 89.0iT - 6.85e3T^{2} \)
23 \( 1 - 120. iT - 1.21e4T^{2} \)
29 \( 1 + 24T + 2.43e4T^{2} \)
31 \( 1 + 70T + 2.97e4T^{2} \)
37 \( 1 - 182T + 5.06e4T^{2} \)
41 \( 1 - 294T + 6.89e4T^{2} \)
43 \( 1 + 4.24iT - 7.95e4T^{2} \)
47 \( 1 + 108. iT - 1.03e5T^{2} \)
53 \( 1 + 147. iT - 1.48e5T^{2} \)
59 \( 1 - 514. iT - 2.05e5T^{2} \)
61 \( 1 - 326. iT - 2.26e5T^{2} \)
67 \( 1 + 880T + 3.00e5T^{2} \)
71 \( 1 + 337. iT - 3.57e5T^{2} \)
73 \( 1 - 178. iT - 3.89e5T^{2} \)
79 \( 1 - 772. iT - 4.93e5T^{2} \)
83 \( 1 - 1.21e3T + 5.71e5T^{2} \)
89 \( 1 + 1.53e3iT - 7.04e5T^{2} \)
97 \( 1 + 196T + 9.12e5T^{2} \)
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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

−13.24275859997159811606916457759, −12.34009633039195019149418368827, −11.62531572229395898878488262486, −9.655604908610199739306963188426, −8.960917921763530838394495914368, −7.44749897744490840866445917875, −5.75453061583741402606114379149, −4.67092600195408856889309482277, −3.84371975304771202768784668169, −1.06699964224165067041201041074, 2.80272473758230927183061136704, 3.71157769955466952845691738512, 5.68665902939013417695247495104, 6.32921320818130283187633644727, 7.88795754273154221829542236793, 9.457393683404052959799211672698, 10.64064200960985064678768025605, 11.81906568912487640316304289130, 12.52276122825959233547411034542, 13.99933741374808131924415334162

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