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Properties

Label 2-99-9.4-c5-0-32
Degree $2$
Conductor $99$
Sign $0.951 - 0.307i$
Analytic cond. $15.8779$
Root an. cond. $3.98472$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.102 + 0.176i)2-s + (11.7 + 10.2i)3-s + (15.9 − 27.6i)4-s + (9.22 − 15.9i)5-s + (−0.609 + 3.12i)6-s + (66.9 + 115. i)7-s + 13.0·8-s + (33.4 + 240. i)9-s + 3.76·10-s + (60.5 + 104. i)11-s + (471. − 161. i)12-s + (242. − 420. i)13-s + (−13.6 + 23.6i)14-s + (272. − 93.4i)15-s + (−509. − 883. i)16-s + 266.·17-s + ⋯
L(s)  = 1  + (0.0180 + 0.0312i)2-s + (0.754 + 0.656i)3-s + (0.499 − 0.864i)4-s + (0.165 − 0.285i)5-s + (−0.00691 + 0.0354i)6-s + (0.516 + 0.894i)7-s + 0.0721·8-s + (0.137 + 0.990i)9-s + 0.0119·10-s + (0.150 + 0.261i)11-s + (0.944 − 0.324i)12-s + (0.398 − 0.689i)13-s + (−0.0186 + 0.0322i)14-s + (0.312 − 0.107i)15-s + (−0.498 − 0.862i)16-s + 0.223·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.951 - 0.307i$
Analytic conductor: \(15.8779\)
Root analytic conductor: \(3.98472\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :5/2),\ 0.951 - 0.307i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.96470 + 0.467190i\)
\(L(\frac12)\) \(\approx\) \(2.96470 + 0.467190i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-11.7 - 10.2i)T \)
11 \( 1 + (-60.5 - 104. i)T \)
good2 \( 1 + (-0.102 - 0.176i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (-9.22 + 15.9i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-66.9 - 115. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
13 \( 1 + (-242. + 420. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 - 266.T + 1.41e6T^{2} \)
19 \( 1 - 1.64e3T + 2.47e6T^{2} \)
23 \( 1 + (-372. + 645. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (524. + 908. i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (1.61e3 - 2.80e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 1.30e4T + 6.93e7T^{2} \)
41 \( 1 + (1.69e3 - 2.94e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (5.83e3 + 1.01e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (320. + 554. i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 1.93e4T + 4.18e8T^{2} \)
59 \( 1 + (1.96e4 - 3.40e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (9.66e3 + 1.67e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-5.93e3 + 1.02e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 7.37e4T + 1.80e9T^{2} \)
73 \( 1 - 1.04e4T + 2.07e9T^{2} \)
79 \( 1 + (1.57e4 + 2.72e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (5.24e4 + 9.08e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 2.23e4T + 5.58e9T^{2} \)
97 \( 1 + (1.66e4 + 2.88e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−13.19013701944177854009608499892, −11.75504993798465151368953082096, −10.72148347116249022629293442093, −9.679786245028251335631038099847, −8.808013932604994940650942939120, −7.53478045216967345164396574367, −5.76779857650782806607377959845, −4.87676704157288160368396528393, −2.94259340324363765418413777965, −1.52310296009845799002842973021, 1.36544732437832175745667434880, 2.89295074206536856132893215991, 4.09202795003830282534168206019, 6.42295356385810901239459495986, 7.40343570580004506121709450350, 8.161859516776041166780729647102, 9.447751950378537225761641938992, 11.00468342613009786080034869925, 11.85980926551788219679668995360, 13.03638429533213423341057272848

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