Properties

Label 2-99-11.9-c1-0-3
Degree $2$
Conductor $99$
Sign $-0.605 + 0.795i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 2.48i)2-s + (−3.92 − 2.85i)4-s + (0.190 + 0.587i)5-s + (0.809 + 0.587i)7-s + (−6.04 + 4.39i)8-s + 1.61·10-s + (3.30 + 0.224i)11-s + (0.0729 − 0.224i)13-s + (2.11 − 1.53i)14-s + (3.04 + 9.37i)16-s + (0.354 + 1.08i)17-s + (−4.73 + 3.44i)19-s + (0.927 − 2.85i)20-s + (3.23 − 8.05i)22-s − 0.236·23-s + ⋯
L(s)  = 1  + (0.572 − 1.76i)2-s + (−1.96 − 1.42i)4-s + (0.0854 + 0.262i)5-s + (0.305 + 0.222i)7-s + (−2.13 + 1.55i)8-s + 0.511·10-s + (0.997 + 0.0676i)11-s + (0.0202 − 0.0622i)13-s + (0.566 − 0.411i)14-s + (0.761 + 2.34i)16-s + (0.0858 + 0.264i)17-s + (−1.08 + 0.789i)19-s + (0.207 − 0.637i)20-s + (0.689 − 1.71i)22-s − 0.0492·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.541549 - 1.09237i\)
\(L(\frac12)\) \(\approx\) \(0.541549 - 1.09237i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.30 - 0.224i)T \)
good2 \( 1 + (-0.809 + 2.48i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.190 - 0.587i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.0729 + 0.224i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.354 - 1.08i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 0.236T + 23T^{2} \)
29 \( 1 + (4.85 + 3.52i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.88 - 5.79i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.04 - 3.66i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.190 + 0.138i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.70T + 43T^{2} \)
47 \( 1 + (8.16 - 5.93i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.118 + 0.363i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.97 - 4.33i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.57 + 10.9i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 1.85T + 67T^{2} \)
71 \( 1 + (3.19 + 9.82i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.61 - 3.35i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.39 + 10.4i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.454 + 1.40i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.23T + 89T^{2} \)
97 \( 1 + (-2.42 + 7.46i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.21903382862828714841697728728, −12.34212606160480480369532205912, −11.49579213424119554016335252283, −10.58914308488668776124766034680, −9.621233081045285182019871440769, −8.465909557683005444294389198268, −6.25377628042621661693132216485, −4.71167265570412505244281419165, −3.48286501579617997147708652733, −1.83112140241498648073084958194, 3.95878818678210987395774707797, 5.09379503607462220272508286657, 6.36893665835094932040845400721, 7.30266765817157788565595770418, 8.528606043985546045880646587795, 9.374689771408453280143777306937, 11.30104352816801899672395321546, 12.74354840792689086502648108786, 13.46890748198337438140272564225, 14.62401658294101409941727026059

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