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Properties

Label 2-99-33.8-c1-0-2
Degree $2$
Conductor $99$
Sign $0.997 - 0.0632i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.205 − 0.149i)2-s + (−0.598 + 1.84i)4-s + (1.71 − 2.36i)5-s + (2.58 + 0.840i)7-s + (0.308 + 0.949i)8-s − 0.741i·10-s + (−2.71 + 1.89i)11-s + (−1.67 − 2.31i)13-s + (0.655 − 0.213i)14-s + (−2.92 − 2.12i)16-s + (−3.60 − 2.62i)17-s + (−1.81 + 0.590i)19-s + (3.32 + 4.57i)20-s + (−0.275 + 0.795i)22-s − 0.816i·23-s + ⋯
L(s)  = 1  + (0.145 − 0.105i)2-s + (−0.299 + 0.920i)4-s + (0.768 − 1.05i)5-s + (0.977 + 0.317i)7-s + (0.109 + 0.335i)8-s − 0.234i·10-s + (−0.820 + 0.572i)11-s + (−0.465 − 0.640i)13-s + (0.175 − 0.0569i)14-s + (−0.731 − 0.531i)16-s + (−0.875 − 0.636i)17-s + (−0.416 + 0.135i)19-s + (0.743 + 1.02i)20-s + (−0.0586 + 0.169i)22-s − 0.170i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.997 - 0.0632i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.997 - 0.0632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13449 + 0.0359067i\)
\(L(\frac12)\) \(\approx\) \(1.13449 + 0.0359067i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.71 - 1.89i)T \)
good2 \( 1 + (-0.205 + 0.149i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.71 + 2.36i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-2.58 - 0.840i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.67 + 2.31i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.60 + 2.62i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.81 - 0.590i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.816iT - 23T^{2} \)
29 \( 1 + (2.95 - 9.07i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.84 + 3.51i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.83 + 5.65i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.60 + 8.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 11.8iT - 43T^{2} \)
47 \( 1 + (-7.34 + 2.38i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.14 - 8.45i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.0887 + 0.0288i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.26 + 1.73i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 1.40T + 67T^{2} \)
71 \( 1 + (-2.12 + 2.91i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.67 + 0.543i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.19 + 5.77i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.33 - 4.60i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.06iT - 89T^{2} \)
97 \( 1 + (1.35 - 0.981i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.60448273602105997128773551108, −12.84935458327847124112529258683, −12.13992399877531191665699003037, −10.81191926003411561892127758327, −9.331328617480702116474669152598, −8.481973566849011586245411040831, −7.44520019333582558720894500308, −5.34329451575505172005784510638, −4.58799818242106153691830156111, −2.34669555998390238218662787717, 2.18107314573745517132346405538, 4.51016408970179449378455678345, 5.83218387833088925905933303348, 6.85736439076616270756905614218, 8.405054767788682948829771180534, 9.874320168063832129558362222830, 10.61536928928313143247234704620, 11.43394147594959106810814049165, 13.38758136589249734555913355453, 13.89175682278449367931997180672

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