Properties

Label 2-99-11.10-c2-0-8
Degree $2$
Conductor $99$
Sign $-0.612 - 0.790i$
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  − 3.53i·2-s − 8.46·4-s − 6.19·5-s + 2.58i·7-s + 15.7i·8-s + 21.8i·10-s + (−6.73 − 8.69i)11-s − 23.7i·13-s + 9.12·14-s + 21.7·16-s + 12.2i·17-s − 3.27i·19-s + 52.4·20-s + (−30.7 + 23.7i)22-s + 14.3·23-s + ⋯
L(s)  = 1  − 1.76i·2-s − 2.11·4-s − 1.23·5-s + 0.369i·7-s + 1.97i·8-s + 2.18i·10-s + (−0.612 − 0.790i)11-s − 1.82i·13-s + 0.651·14-s + 1.36·16-s + 0.719i·17-s − 0.172i·19-s + 2.62·20-s + (−1.39 + 1.08i)22-s + 0.623·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.612 - 0.790i$
Analytic conductor: \(2.69755\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1),\ -0.612 - 0.790i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.253004 + 0.515702i\)
\(L(\frac12)\) \(\approx\) \(0.253004 + 0.515702i\)
\(L(2)\) 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 \( 1 + (6.73 + 8.69i)T \)
good2 \( 1 + 3.53iT - 4T^{2} \)
5 \( 1 + 6.19T + 25T^{2} \)
7 \( 1 - 2.58iT - 49T^{2} \)
13 \( 1 + 23.7iT - 169T^{2} \)
17 \( 1 - 12.2iT - 289T^{2} \)
19 \( 1 + 3.27iT - 361T^{2} \)
23 \( 1 - 14.3T + 529T^{2} \)
29 \( 1 + 38.5iT - 841T^{2} \)
31 \( 1 + 11.1T + 961T^{2} \)
37 \( 1 + 12.5T + 1.36e3T^{2} \)
41 \( 1 + 1.38iT - 1.68e3T^{2} \)
43 \( 1 + 23.9iT - 1.84e3T^{2} \)
47 \( 1 + 19.8T + 2.20e3T^{2} \)
53 \( 1 - 12.0T + 2.80e3T^{2} \)
59 \( 1 - 62.7T + 3.48e3T^{2} \)
61 \( 1 - 21.3iT - 3.72e3T^{2} \)
67 \( 1 + 34T + 4.48e3T^{2} \)
71 \( 1 - 69.2T + 5.04e3T^{2} \)
73 \( 1 - 39.9iT - 5.32e3T^{2} \)
79 \( 1 + 97.6iT - 6.24e3T^{2} \)
83 \( 1 + 71.9iT - 6.88e3T^{2} \)
89 \( 1 + 107.T + 7.92e3T^{2} \)
97 \( 1 + 166.T + 9.40e3T^{2} \)
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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

−12.71692555950177422458913399980, −11.81257913234195749655003323529, −10.95298681321864520106781771119, −10.20731013450066108123096572831, −8.682455913229222244646871275212, −7.88008278307384103051188761633, −5.42064793063244208870924899004, −3.86702034632014732685982197092, −2.83631151900026662801285429690, −0.44472646266925324372337709955, 4.08420637090353443881296211239, 5.03960608206506129436887541313, 6.88145112202815231241828686318, 7.31824401889604623559207793987, 8.475701547038382132766080744380, 9.526784787038551729816924164938, 11.23584574537176635156005456843, 12.46060122650998994524639631909, 13.73998403148635193034467838042, 14.59320987962549858920434904606

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