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Properties

Label 2-99-11.3-c3-0-5
Degree $2$
Conductor $99$
Sign $0.799 - 0.601i$
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  + (2.02 − 1.47i)2-s + (−0.539 + 1.65i)4-s + (8.44 + 6.13i)5-s + (−10.1 + 31.1i)7-s + (7.53 + 23.1i)8-s + 26.0·10-s + (−12.6 − 34.2i)11-s + (59.2 − 43.0i)13-s + (25.3 + 77.9i)14-s + (38.0 + 27.6i)16-s + (−44.7 − 32.4i)17-s + (27.9 + 86.1i)19-s + (−14.7 + 10.7i)20-s + (−75.9 − 50.6i)22-s + 91.1·23-s + ⋯
L(s)  = 1  + (0.715 − 0.519i)2-s + (−0.0673 + 0.207i)4-s + (0.755 + 0.548i)5-s + (−0.546 + 1.68i)7-s + (0.332 + 1.02i)8-s + 0.825·10-s + (−0.347 − 0.937i)11-s + (1.26 − 0.918i)13-s + (0.483 + 1.48i)14-s + (0.594 + 0.431i)16-s + (−0.638 − 0.463i)17-s + (0.337 + 1.03i)19-s + (−0.164 + 0.119i)20-s + (−0.735 − 0.490i)22-s + 0.826·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.799 - 0.601i$
Analytic conductor: \(5.84118\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :3/2),\ 0.799 - 0.601i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.15874 + 0.721334i\)
\(L(\frac12)\) \(\approx\) \(2.15874 + 0.721334i\)
\(L(\frac{5}{2})\) 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 \( 1 + (12.6 + 34.2i)T \)
good2 \( 1 + (-2.02 + 1.47i)T + (2.47 - 7.60i)T^{2} \)
5 \( 1 + (-8.44 - 6.13i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (10.1 - 31.1i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (-59.2 + 43.0i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (44.7 + 32.4i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (-27.9 - 86.1i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 - 91.1T + 1.21e4T^{2} \)
29 \( 1 + (-23.7 + 72.9i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (18.6 - 13.5i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-38.8 + 119. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (43.2 + 133. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 146.T + 7.95e4T^{2} \)
47 \( 1 + (-68.4 - 210. i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (35.5 - 25.8i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-124. + 382. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (-328. - 238. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 - 221.T + 3.00e5T^{2} \)
71 \( 1 + (606. + 440. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (68.6 - 211. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (-954. + 693. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (-547. - 397. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + (-198. + 144. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−13.35746871770884209090120016763, −12.62730941920786277353682562438, −11.54294118352035387803831362235, −10.59856481833726246772898100477, −9.114556944734518886344030210227, −8.195974392265541514648924148489, −6.11556621083527760370771879447, −5.47007964405789307060466679550, −3.35838866696270854771252627182, −2.45587868625075226340583282751, 1.20131091563991210682145979013, 3.95851926740252684008404696909, 4.94752964693075441507158566756, 6.44762196582114766891085403006, 7.15662683948269981972445487104, 9.100454063277398739012881570578, 10.03685327705436819091885112398, 11.01746706837894387447671736307, 13.02075040841387267489330435608, 13.31098256692418094469206799640

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