Properties

Label 2-99-33.14-c2-0-4
Degree $2$
Conductor $99$
Sign $0.909 + 0.415i$
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  + (0.480 + 0.660i)2-s + (1.02 − 3.16i)4-s + (0.615 − 0.847i)5-s + (2.11 − 6.49i)7-s + (5.69 − 1.85i)8-s + 0.855·10-s + (2.57 + 10.6i)11-s + (6.23 − 4.52i)13-s + (5.30 − 1.72i)14-s + (−6.82 − 4.96i)16-s + (−5.52 + 7.59i)17-s + (2.91 + 8.95i)19-s + (−2.05 − 2.82i)20-s + (−5.83 + 6.83i)22-s − 4.82i·23-s + ⋯
L(s)  = 1  + (0.240 + 0.330i)2-s + (0.257 − 0.792i)4-s + (0.123 − 0.169i)5-s + (0.301 − 0.927i)7-s + (0.711 − 0.231i)8-s + 0.0855·10-s + (0.233 + 0.972i)11-s + (0.479 − 0.348i)13-s + (0.378 − 0.123i)14-s + (−0.426 − 0.310i)16-s + (−0.324 + 0.447i)17-s + (0.153 + 0.471i)19-s + (−0.102 − 0.141i)20-s + (−0.265 + 0.310i)22-s − 0.209i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.909 + 0.415i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ 0.909 + 0.415i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.61931 - 0.352753i\)
\(L(\frac12)\) \(\approx\) \(1.61931 - 0.352753i\)
\(L(2)\) 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 \( 1 + (-2.57 - 10.6i)T \)
good2 \( 1 + (-0.480 - 0.660i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (-0.615 + 0.847i)T + (-7.72 - 23.7i)T^{2} \)
7 \( 1 + (-2.11 + 6.49i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (-6.23 + 4.52i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (5.52 - 7.59i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-2.91 - 8.95i)T + (-292. + 212. i)T^{2} \)
23 \( 1 + 4.82iT - 529T^{2} \)
29 \( 1 + (44.4 + 14.4i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (15.3 - 11.1i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (14.5 - 44.8i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-7.22 + 2.34i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 50.6T + 1.84e3T^{2} \)
47 \( 1 + (62.9 - 20.4i)T + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-22.5 - 31.0i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (-111. - 36.2i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (65.8 + 47.8i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 70.1T + 4.48e3T^{2} \)
71 \( 1 + (-25.4 + 35.0i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-14.1 + 43.6i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-69.3 + 50.4i)T + (1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (93.5 - 128. i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 154. iT - 7.92e3T^{2} \)
97 \( 1 + (124. - 90.2i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−13.70499494226725666090452680360, −12.78482729487374850400681656258, −11.25286315138303733638743829067, −10.42120559110195228849606499424, −9.401128095941143134299894726544, −7.73213988468879532303826054251, −6.72157576531602023522189697949, −5.40256558705478333297464704462, −4.11887729480887480595529663965, −1.52738880836153995447445703315, 2.34734929376908244368342398248, 3.78853046455184594520427985215, 5.50426505052499709775746694275, 6.94012814530396042912456758302, 8.319793813669256732939098187987, 9.183701721401080142887965337899, 11.01273709525752925206292297269, 11.52139404530117734951076592350, 12.62681945924802983752740538192, 13.57711667330503357842650092213

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