Properties

Label 2-99-33.29-c7-0-6
Degree $2$
Conductor $99$
Sign $0.917 - 0.398i$
Analytic cond. $30.9261$
Root an. cond. $5.56112$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−14.7 − 10.7i)2-s + (63.6 + 196. i)4-s + (194. + 268. i)5-s + (−72.5 + 23.5i)7-s + (441. − 1.35e3i)8-s − 6.06e3i·10-s + (4.37e3 + 597. i)11-s + (5.50e3 − 7.57e3i)13-s + (1.32e3 + 431. i)14-s + (229. − 166. i)16-s + (−9.94e3 + 7.22e3i)17-s + (−2.06e4 − 6.69e3i)19-s + (−4.01e4 + 5.52e4i)20-s + (−5.82e4 − 5.58e4i)22-s + 8.83e4i·23-s + ⋯
L(s)  = 1  + (−1.30 − 0.949i)2-s + (0.497 + 1.53i)4-s + (0.697 + 0.959i)5-s + (−0.0799 + 0.0259i)7-s + (0.304 − 0.937i)8-s − 1.91i·10-s + (0.990 + 0.135i)11-s + (0.695 − 0.956i)13-s + (0.129 + 0.0419i)14-s + (0.0139 − 0.0101i)16-s + (−0.491 + 0.356i)17-s + (−0.689 − 0.224i)19-s + (−1.12 + 1.54i)20-s + (−1.16 − 1.11i)22-s + 1.51i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.917 - 0.398i$
Analytic conductor: \(30.9261\)
Root analytic conductor: \(5.56112\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :7/2),\ 0.917 - 0.398i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.03747 + 0.215831i\)
\(L(\frac12)\) \(\approx\) \(1.03747 + 0.215831i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-4.37e3 - 597. i)T \)
good2 \( 1 + (14.7 + 10.7i)T + (39.5 + 121. i)T^{2} \)
5 \( 1 + (-194. - 268. i)T + (-2.41e4 + 7.43e4i)T^{2} \)
7 \( 1 + (72.5 - 23.5i)T + (6.66e5 - 4.84e5i)T^{2} \)
13 \( 1 + (-5.50e3 + 7.57e3i)T + (-1.93e7 - 5.96e7i)T^{2} \)
17 \( 1 + (9.94e3 - 7.22e3i)T + (1.26e8 - 3.90e8i)T^{2} \)
19 \( 1 + (2.06e4 + 6.69e3i)T + (7.23e8 + 5.25e8i)T^{2} \)
23 \( 1 - 8.83e4iT - 3.40e9T^{2} \)
29 \( 1 + (-2.10e4 - 6.49e4i)T + (-1.39e10 + 1.01e10i)T^{2} \)
31 \( 1 + (-3.95e4 - 2.87e4i)T + (8.50e9 + 2.61e10i)T^{2} \)
37 \( 1 + (-1.23e4 - 3.80e4i)T + (-7.68e10 + 5.57e10i)T^{2} \)
41 \( 1 + (-1.29e5 + 3.98e5i)T + (-1.57e11 - 1.14e11i)T^{2} \)
43 \( 1 + 2.73e5iT - 2.71e11T^{2} \)
47 \( 1 + (-1.12e6 - 3.64e5i)T + (4.09e11 + 2.97e11i)T^{2} \)
53 \( 1 + (-4.31e5 + 5.93e5i)T + (-3.63e11 - 1.11e12i)T^{2} \)
59 \( 1 + (1.86e6 - 6.07e5i)T + (2.01e12 - 1.46e12i)T^{2} \)
61 \( 1 + (-1.39e6 - 1.92e6i)T + (-9.71e11 + 2.98e12i)T^{2} \)
67 \( 1 + 3.69e5T + 6.06e12T^{2} \)
71 \( 1 + (-2.08e6 - 2.87e6i)T + (-2.81e12 + 8.64e12i)T^{2} \)
73 \( 1 + (1.18e6 - 3.84e5i)T + (8.93e12 - 6.49e12i)T^{2} \)
79 \( 1 + (-2.54e6 + 3.49e6i)T + (-5.93e12 - 1.82e13i)T^{2} \)
83 \( 1 + (2.09e6 - 1.52e6i)T + (8.38e12 - 2.58e13i)T^{2} \)
89 \( 1 - 6.18e6iT - 4.42e13T^{2} \)
97 \( 1 + (7.39e6 + 5.36e6i)T + (2.49e13 + 7.68e13i)T^{2} \)
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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

−12.19441256998772485596940331743, −11.04052432506348896107813320077, −10.48113171889955032836632178145, −9.475830190212821631951036765776, −8.562166371463974510125976104681, −7.19362764592201718644148913015, −5.94922594708138035523369405861, −3.56759402452966582302980636583, −2.34958557231595359393040809858, −1.13921469085353107772165168731, 0.60238242627942358778954330819, 1.74539803672714139582931241169, 4.39475394463238400410391651615, 6.05632058229534654923178297904, 6.73990985480184510619882377284, 8.342976174797122116901645150730, 8.995158309707727933581819793409, 9.716144094431894114102457244798, 11.02766516888299354874246184821, 12.44977162523734307921615285484

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