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Properties

Label 2-99-33.26-c2-0-0
Degree $2$
Conductor $99$
Sign $-0.961 - 0.273i$
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  + (−1.75 + 2.41i)2-s + (−1.50 − 4.64i)4-s + (4.61 + 6.35i)5-s + (0.867 + 2.67i)7-s + (2.50 + 0.812i)8-s − 23.4·10-s + (−8.89 + 6.47i)11-s + (1.22 + 0.892i)13-s + (−7.95 − 2.58i)14-s + (9.45 − 6.87i)16-s + (−15.5 − 21.3i)17-s + (−4.49 + 13.8i)19-s + (22.5 − 31.0i)20-s + (−0.0177 − 32.7i)22-s + 25.0i·23-s + ⋯
L(s)  = 1  + (−0.875 + 1.20i)2-s + (−0.377 − 1.16i)4-s + (0.923 + 1.27i)5-s + (0.123 + 0.381i)7-s + (0.312 + 0.101i)8-s − 2.34·10-s + (−0.808 + 0.588i)11-s + (0.0944 + 0.0686i)13-s + (−0.568 − 0.184i)14-s + (0.591 − 0.429i)16-s + (−0.912 − 1.25i)17-s + (−0.236 + 0.727i)19-s + (1.12 − 1.55i)20-s + (−0.000808 − 1.49i)22-s + 1.09i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.273i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.961 - 0.273i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ -0.961 - 0.273i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.117001 + 0.839354i\)
\(L(\frac12)\) \(\approx\) \(0.117001 + 0.839354i\)
\(L(2)\) 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 \( 1 + (8.89 - 6.47i)T \)
good2 \( 1 + (1.75 - 2.41i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (-4.61 - 6.35i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (-0.867 - 2.67i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (-1.22 - 0.892i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (15.5 + 21.3i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (4.49 - 13.8i)T + (-292. - 212. i)T^{2} \)
23 \( 1 - 25.0iT - 529T^{2} \)
29 \( 1 + (-48.2 + 15.6i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-41.7 - 30.3i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (6.04 + 18.5i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (29.4 + 9.55i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 2.11T + 1.84e3T^{2} \)
47 \( 1 + (-38.2 - 12.4i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-24.1 + 33.2i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (-91.3 + 29.6i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-2.34 + 1.70i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 88.2T + 4.48e3T^{2} \)
71 \( 1 + (-6.65 - 9.16i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-32.3 - 99.6i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (13.5 + 9.86i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (13.3 + 18.4i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 30.7iT - 7.92e3T^{2} \)
97 \( 1 + (-23.0 - 16.7i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−14.35082013939277566127508233346, −13.57658622211125927247882181141, −11.89422294769816789087689298762, −10.39808669871419616551239876776, −9.762812586205907600980504873862, −8.527866527898879977418577119054, −7.25326535183051687351785303661, −6.50505186738636909888185026469, −5.33497061095773050756471752324, −2.60300572984214637555062511547, 0.882015120407840685948333804164, 2.44671551451196568857797434332, 4.56683041428183464306847551022, 6.12687948840924604432567742504, 8.396856065047993877165395464906, 8.783800866723132585395707027397, 10.13923330541389379567645437557, 10.70072564027156317266677286427, 12.05512896090942380923843948557, 13.00799137103370618412381572109

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