Properties

Label 2-99-3.2-c2-0-1
Degree $2$
Conductor $99$
Sign $-0.816 - 0.577i$
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  + 2.48i·2-s − 2.17·4-s + 4.68i·5-s − 3.30·7-s + 4.53i·8-s − 11.6·10-s + 3.31i·11-s + 1.00·13-s − 8.20i·14-s − 19.9·16-s − 6.45i·17-s + 25.1·19-s − 10.2i·20-s − 8.24·22-s − 24.4i·23-s + ⋯
L(s)  = 1  + 1.24i·2-s − 0.543·4-s + 0.937i·5-s − 0.471·7-s + 0.566i·8-s − 1.16·10-s + 0.301i·11-s + 0.0775·13-s − 0.586i·14-s − 1.24·16-s − 0.379i·17-s + 1.32·19-s − 0.510i·20-s − 0.374·22-s − 1.06i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ -0.816 - 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.390064 + 1.22724i\)
\(L(\frac12)\) \(\approx\) \(0.390064 + 1.22724i\)
\(L(2)\) 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 \( 1 - 3.31iT \)
good2 \( 1 - 2.48iT - 4T^{2} \)
5 \( 1 - 4.68iT - 25T^{2} \)
7 \( 1 + 3.30T + 49T^{2} \)
13 \( 1 - 1.00T + 169T^{2} \)
17 \( 1 + 6.45iT - 289T^{2} \)
19 \( 1 - 25.1T + 361T^{2} \)
23 \( 1 + 24.4iT - 529T^{2} \)
29 \( 1 - 50.8iT - 841T^{2} \)
31 \( 1 - 48.4T + 961T^{2} \)
37 \( 1 - 65.8T + 1.36e3T^{2} \)
41 \( 1 + 6.41iT - 1.68e3T^{2} \)
43 \( 1 + 48.5T + 1.84e3T^{2} \)
47 \( 1 + 56.8iT - 2.20e3T^{2} \)
53 \( 1 + 67.3iT - 2.80e3T^{2} \)
59 \( 1 + 0.307iT - 3.48e3T^{2} \)
61 \( 1 + 86.8T + 3.72e3T^{2} \)
67 \( 1 - 29.6T + 4.48e3T^{2} \)
71 \( 1 + 40.9iT - 5.04e3T^{2} \)
73 \( 1 + 61.0T + 5.32e3T^{2} \)
79 \( 1 - 85.4T + 6.24e3T^{2} \)
83 \( 1 - 64.7iT - 6.88e3T^{2} \)
89 \( 1 - 64.1iT - 7.92e3T^{2} \)
97 \( 1 + 86.8T + 9.40e3T^{2} \)
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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

−14.41170906103071854149990874096, −13.43495092433614576792021441009, −11.96412904377732011569523422779, −10.81935994455356799816932806348, −9.614846100951381709261089665161, −8.235098509650900591324022774011, −7.06047022681368359049627493281, −6.43582175903541550346058559994, −5.00896290831815675914722534385, −2.93963427071291783872155317884, 1.09294384407139854265682891105, 3.01584346191154152023953725264, 4.46461980900513337316974775279, 6.12239081148757597872577383111, 7.86453986965781091152509169432, 9.307547316971332565776294069646, 9.960301561335466825374009749287, 11.34397042325621815859246404651, 12.03585108046944318793517266585, 13.06316974132319103228647917381

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