Properties

Label 2-99-11.8-c2-0-3
Degree $2$
Conductor $99$
Sign $-0.220 - 0.975i$
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  + (1.80 + 2.48i)2-s + (−1.69 + 5.20i)4-s + (3.23 + 2.35i)5-s + (−0.854 − 0.277i)7-s + (−4.30 + 1.40i)8-s + 12.3i·10-s + (−10.8 + 1.67i)11-s + (−5 − 6.88i)13-s + (−0.854 − 2.62i)14-s + (6.42 + 4.66i)16-s + (14.5 − 20.0i)17-s + (11.2 − 3.66i)19-s + (−17.7 + 12.8i)20-s + (−23.8 − 24.0i)22-s + 7.23·23-s + ⋯
L(s)  = 1  + (0.904 + 1.24i)2-s + (−0.422 + 1.30i)4-s + (0.647 + 0.470i)5-s + (−0.122 − 0.0396i)7-s + (−0.538 + 0.175i)8-s + 1.23i·10-s + (−0.988 + 0.152i)11-s + (−0.384 − 0.529i)13-s + (−0.0610 − 0.187i)14-s + (0.401 + 0.291i)16-s + (0.854 − 1.17i)17-s + (0.593 − 0.192i)19-s + (−0.885 + 0.643i)20-s + (−1.08 − 1.09i)22-s + 0.314·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.220 - 0.975i$
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: \((2,\ 99,\ (\ :1),\ -0.220 - 0.975i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.33257 + 1.66798i\)
\(L(\frac12)\) \(\approx\) \(1.33257 + 1.66798i\)
\(L(2)\) 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 \( 1 + (10.8 - 1.67i)T \)
good2 \( 1 + (-1.80 - 2.48i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (-3.23 - 2.35i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (0.854 + 0.277i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (5 + 6.88i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (-14.5 + 20.0i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-11.2 + 3.66i)T + (292. - 212. i)T^{2} \)
23 \( 1 - 7.23T + 529T^{2} \)
29 \( 1 + (-3.29 - 1.06i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (26.7 - 19.4i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-12.4 + 38.2i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (1.24 - 0.403i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 33.0iT - 1.84e3T^{2} \)
47 \( 1 + (7.03 + 21.6i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (63.5 - 46.1i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (9.66 - 29.7i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (16.5 - 22.7i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 76.5T + 4.48e3T^{2} \)
71 \( 1 + (50.4 + 36.6i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-89.5 - 29.0i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-38.7 - 53.3i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (33.3 - 45.8i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 62.2T + 7.92e3T^{2} \)
97 \( 1 + (58.5 - 42.5i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−14.08739050996393998526976790028, −13.28961050512169790650234246457, −12.32681099229582438818423066625, −10.67874275075683546028755532021, −9.600623639454218331680525822802, −7.86208297219498821027088470108, −7.06383149669678687868769968854, −5.76992838221679031801503041478, −4.94340296374569361003932209293, −3.01650031161770013409955576841, 1.73612757667174877738344866303, 3.29360658511120770333031602520, 4.84779648901546311985560239426, 5.84168107808813450883461116481, 7.84194993958513371589912902940, 9.478187163293610985764691510454, 10.32869945961975639359311238390, 11.38569573412292121873336776484, 12.51195060785668633618949744546, 13.10021035991964999543296557659

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