Properties

Label 2-99-3.2-c2-0-5
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  + 0.192i·2-s + 3.96·4-s − 7.62i·5-s − 2.45·7-s + 1.53i·8-s + 1.46·10-s − 3.31i·11-s + 20.4·13-s − 0.473i·14-s + 15.5·16-s + 14.2i·17-s − 25.2·19-s − 30.2i·20-s + 0.638·22-s + 7.64i·23-s + ⋯
L(s)  = 1  + 0.0963i·2-s + 0.990·4-s − 1.52i·5-s − 0.350·7-s + 0.191i·8-s + 0.146·10-s − 0.301i·11-s + 1.57·13-s − 0.0338i·14-s + 0.972·16-s + 0.841i·17-s − 1.32·19-s − 1.51i·20-s + 0.0290·22-s + 0.332i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.50460 - 0.478220i\)
\(L(\frac12)\) \(\approx\) \(1.50460 - 0.478220i\)
\(L(2)\) 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 \( 1 - 0.192iT - 4T^{2} \)
5 \( 1 + 7.62iT - 25T^{2} \)
7 \( 1 + 2.45T + 49T^{2} \)
13 \( 1 - 20.4T + 169T^{2} \)
17 \( 1 - 14.2iT - 289T^{2} \)
19 \( 1 + 25.2T + 361T^{2} \)
23 \( 1 - 7.64iT - 529T^{2} \)
29 \( 1 - 43.4iT - 841T^{2} \)
31 \( 1 + 33.4T + 961T^{2} \)
37 \( 1 - 24.5T + 1.36e3T^{2} \)
41 \( 1 - 21.8iT - 1.68e3T^{2} \)
43 \( 1 + 17.2T + 1.84e3T^{2} \)
47 \( 1 + 76.8iT - 2.20e3T^{2} \)
53 \( 1 - 3.42iT - 2.80e3T^{2} \)
59 \( 1 - 68.7iT - 3.48e3T^{2} \)
61 \( 1 - 31.6T + 3.72e3T^{2} \)
67 \( 1 - 30.0T + 4.48e3T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 - 131.T + 5.32e3T^{2} \)
79 \( 1 + 10.6T + 6.24e3T^{2} \)
83 \( 1 + 126. iT - 6.88e3T^{2} \)
89 \( 1 + 114. iT - 7.92e3T^{2} \)
97 \( 1 + 109.T + 9.40e3T^{2} \)
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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

−13.15951125683423259193436294684, −12.69616020941054263230023674858, −11.44975474641831683133811760732, −10.49242801914829279759395810245, −8.899587468459361129912280467307, −8.212133449231634621973819424220, −6.54989642386214829361046822516, −5.51666761476179908773503919558, −3.76791304914603876587605377418, −1.50825432679160940571810485086, 2.37018323971967355982322231806, 3.64566846317129982146168578188, 6.16492224891397776236856513674, 6.73112678231266129649095057373, 7.961001581959298424916835011078, 9.719713642601990320223996199029, 10.92339468688349341162247648328, 11.19243726455338647234754776822, 12.63354722760201055357540406395, 13.87223171907757204735243815500

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