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Properties

Label 2-99-3.2-c2-0-3
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  − 1.60i·2-s + 1.41·4-s + 6.21i·5-s + 11.1·7-s − 8.70i·8-s + 9.98·10-s − 3.31i·11-s − 17.7·13-s − 17.9i·14-s − 8.31·16-s − 8.53i·17-s + 16.4·19-s + 8.80i·20-s − 5.32·22-s + 25.0i·23-s + ⋯
L(s)  = 1  − 0.803i·2-s + 0.354·4-s + 1.24i·5-s + 1.59·7-s − 1.08i·8-s + 0.998·10-s − 0.301i·11-s − 1.36·13-s − 1.27i·14-s − 0.519·16-s − 0.502i·17-s + 0.866·19-s + 0.440i·20-s − 0.242·22-s + 1.09i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.55885 - 0.495461i\)
\(L(\frac12)\) \(\approx\) \(1.55885 - 0.495461i\)
\(L(2)\) 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 \( 1 + 1.60iT - 4T^{2} \)
5 \( 1 - 6.21iT - 25T^{2} \)
7 \( 1 - 11.1T + 49T^{2} \)
13 \( 1 + 17.7T + 169T^{2} \)
17 \( 1 + 8.53iT - 289T^{2} \)
19 \( 1 - 16.4T + 361T^{2} \)
23 \( 1 - 25.0iT - 529T^{2} \)
29 \( 1 + 9.46iT - 841T^{2} \)
31 \( 1 + 46.1T + 961T^{2} \)
37 \( 1 + 37.4T + 1.36e3T^{2} \)
41 \( 1 - 51.8iT - 1.68e3T^{2} \)
43 \( 1 + 55.6T + 1.84e3T^{2} \)
47 \( 1 + 37.8iT - 2.20e3T^{2} \)
53 \( 1 + 58.5iT - 2.80e3T^{2} \)
59 \( 1 - 44.6iT - 3.48e3T^{2} \)
61 \( 1 + 10.1T + 3.72e3T^{2} \)
67 \( 1 + 58.3T + 4.48e3T^{2} \)
71 \( 1 + 75.1iT - 5.04e3T^{2} \)
73 \( 1 - 112.T + 5.32e3T^{2} \)
79 \( 1 - 52.2T + 6.24e3T^{2} \)
83 \( 1 + 18.0iT - 6.88e3T^{2} \)
89 \( 1 - 136. iT - 7.92e3T^{2} \)
97 \( 1 - 127.T + 9.40e3T^{2} \)
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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

−13.61430173996342180849064753950, −11.97023002149477928839766795631, −11.43874572246185817232559991725, −10.63865487431450717929756649472, −9.607381633568827573141799499822, −7.73487934914287659853159935857, −6.98113036203136479296606225408, −5.16861813201072160803308590498, −3.30269668537560490287510187510, −1.95146832998413090276851987671, 1.85412047241642429588746034994, 4.76445427698419078254017929745, 5.37375796084896131956724437492, 7.20282649013378807408609092704, 8.066783650643853684855015883076, 8.994919424395936775496395071358, 10.63866475758284980246372358087, 11.82797613058110814962638024829, 12.55170884197239881904114022224, 14.18294234559468313230571516208

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