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Properties

Label 2-99-33.32-c3-0-7
Degree $2$
Conductor $99$
Sign $0.969 - 0.246i$
Analytic cond. $5.84118$
Root an. cond. $2.41685$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.00·2-s + 17.0·4-s + 7.07i·5-s + 10.4i·7-s + 45.5·8-s + 35.4i·10-s + (13.0 − 34.0i)11-s − 43.7i·13-s + 52.4i·14-s + 91.2·16-s − 115.·17-s − 89.3i·19-s + 120. i·20-s + (65.4 − 170. i)22-s + 213. i·23-s + ⋯
L(s)  = 1  + 1.77·2-s + 2.13·4-s + 0.632i·5-s + 0.565i·7-s + 2.01·8-s + 1.11i·10-s + (0.358 − 0.933i)11-s − 0.932i·13-s + 1.00i·14-s + 1.42·16-s − 1.64·17-s − 1.07i·19-s + 1.35i·20-s + (0.634 − 1.65i)22-s + 1.93i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.969 - 0.246i$
Analytic conductor: \(5.84118\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :3/2),\ 0.969 - 0.246i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.95176 + 0.495078i\)
\(L(\frac12)\) \(\approx\) \(3.95176 + 0.495078i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-13.0 + 34.0i)T \)
good2 \( 1 - 5.00T + 8T^{2} \)
5 \( 1 - 7.07iT - 125T^{2} \)
7 \( 1 - 10.4iT - 343T^{2} \)
13 \( 1 + 43.7iT - 2.19e3T^{2} \)
17 \( 1 + 115.T + 4.91e3T^{2} \)
19 \( 1 + 89.3iT - 6.85e3T^{2} \)
23 \( 1 - 213. iT - 1.21e4T^{2} \)
29 \( 1 + 124.T + 2.43e4T^{2} \)
31 \( 1 + 149.T + 2.97e4T^{2} \)
37 \( 1 - 161.T + 5.06e4T^{2} \)
41 \( 1 - 172.T + 6.89e4T^{2} \)
43 \( 1 + 258. iT - 7.95e4T^{2} \)
47 \( 1 - 240. iT - 1.03e5T^{2} \)
53 \( 1 - 334. iT - 1.48e5T^{2} \)
59 \( 1 + 382. iT - 2.05e5T^{2} \)
61 \( 1 + 609. iT - 2.26e5T^{2} \)
67 \( 1 - 260T + 3.00e5T^{2} \)
71 \( 1 + 17.3iT - 3.57e5T^{2} \)
73 \( 1 - 787. iT - 3.89e5T^{2} \)
79 \( 1 - 1.00e3iT - 4.93e5T^{2} \)
83 \( 1 + 183.T + 5.71e5T^{2} \)
89 \( 1 - 11.1iT - 7.04e5T^{2} \)
97 \( 1 - 1.79e3T + 9.12e5T^{2} \)
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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

−13.43109678540484231130061826537, −12.74744084975544375697755475771, −11.28427373034384760473713162757, −11.08116275507617065475598517522, −9.075076693417420284466621003871, −7.30730290414058895205055051596, −6.19365053826785475717336994666, −5.23439529866529323825736240340, −3.67522681921224379340901119962, −2.54353184525994104238703694397, 2.08385315222652905755081719439, 4.11072530015319317112594811754, 4.66893560433983426044055522571, 6.26566285730800381610266695383, 7.19078681083784253703996424160, 8.970081034862188775336679799567, 10.58619889079451033624159423792, 11.67525022119339358998788738123, 12.63737688117609310101645363462, 13.23301096541369852590429687882

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