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Properties

Label 2-99-11.8-c2-0-4
Degree $2$
Conductor $99$
Sign $0.866 - 0.498i$
Analytic cond. $2.69755$
Root an. cond. $1.64242$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.816 + 1.12i)2-s + (0.639 − 1.96i)4-s + (3.53 + 2.57i)5-s + (0.582 + 0.189i)7-s + (8.01 − 2.60i)8-s + 6.07i·10-s + (4.81 + 9.89i)11-s + (−5.26 − 7.23i)13-s + (0.263 + 0.809i)14-s + (2.77 + 2.01i)16-s + (−1.74 + 2.40i)17-s + (−9.51 + 3.09i)19-s + (7.32 − 5.32i)20-s + (−7.19 + 13.4i)22-s − 28.6·23-s + ⋯
L(s)  = 1  + (0.408 + 0.561i)2-s + (0.159 − 0.492i)4-s + (0.707 + 0.514i)5-s + (0.0832 + 0.0270i)7-s + (1.00 − 0.325i)8-s + 0.607i·10-s + (0.437 + 0.899i)11-s + (−0.404 − 0.556i)13-s + (0.0187 + 0.0578i)14-s + (0.173 + 0.126i)16-s + (−0.102 + 0.141i)17-s + (−0.500 + 0.162i)19-s + (0.366 − 0.266i)20-s + (−0.326 + 0.612i)22-s − 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.866 - 0.498i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.866 - 0.498i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.81165 + 0.483973i\)
\(L(\frac12)\) \(\approx\) \(1.81165 + 0.483973i\)
\(L(2)\) 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 \( 1 + (-4.81 - 9.89i)T \)
good2 \( 1 + (-0.816 - 1.12i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (-3.53 - 2.57i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (-0.582 - 0.189i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (5.26 + 7.23i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (1.74 - 2.40i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (9.51 - 3.09i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 28.6T + 529T^{2} \)
29 \( 1 + (33.6 + 10.9i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-40.7 + 29.6i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-0.539 + 1.66i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (56.5 - 18.3i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 43.6iT - 1.84e3T^{2} \)
47 \( 1 + (-12.8 - 39.5i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-53.0 + 38.5i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (17.4 - 53.6i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (50.2 - 69.2i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 27.8T + 4.48e3T^{2} \)
71 \( 1 + (-95.3 - 69.2i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-112. - 36.4i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (73.5 + 101. i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-19.2 + 26.4i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 19.9T + 7.92e3T^{2} \)
97 \( 1 + (-141. + 102. i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−13.95875063495560518449599647496, −12.99561626965802696032137983749, −11.64014618041529332666902947534, −10.25467927758757327149920187830, −9.765936553360057396897167357040, −7.889389158630400717976215924281, −6.65185412421318878693427900718, −5.80536319005283502194437240942, −4.39632969318692777558602196457, −2.07594084597955369617442706552, 1.95468184523713383364156042795, 3.68438507580696969899850866652, 5.11158844685682024607154225067, 6.63089966648630462293010382508, 8.144141538355118139904573825061, 9.231101700047059545647743410776, 10.55738131163506713277423908859, 11.65876970884768122446098992962, 12.45109901469880443277499819016, 13.58681732021413763712936531264

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