Properties

Label 2-99-11.9-c1-0-0
Degree $2$
Conductor $99$
Sign $-0.213 - 0.976i$
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  + (−0.476 + 1.46i)2-s + (−0.309 − 0.224i)4-s + (0.476 + 1.46i)5-s + (−0.190 − 0.138i)7-s + (−2.02 + 1.46i)8-s − 2.38·10-s + (2.31 − 2.37i)11-s + (−1.30 + 4.02i)13-s + (0.294 − 0.214i)14-s + (−1.42 − 4.39i)16-s + (−1.83 − 5.65i)17-s + (3.73 − 2.71i)19-s + (0.182 − 0.560i)20-s + (2.38 + 4.53i)22-s + 7.49·23-s + ⋯
L(s)  = 1  + (−0.337 + 1.03i)2-s + (−0.154 − 0.112i)4-s + (0.213 + 0.656i)5-s + (−0.0721 − 0.0524i)7-s + (−0.714 + 0.518i)8-s − 0.753·10-s + (0.698 − 0.716i)11-s + (−0.363 + 1.11i)13-s + (0.0787 − 0.0572i)14-s + (−0.356 − 1.09i)16-s + (−0.445 − 1.37i)17-s + (0.857 − 0.622i)19-s + (0.0407 − 0.125i)20-s + (0.507 + 0.965i)22-s + 1.56·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.213 - 0.976i$
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: \((2,\ 99,\ (\ :1/2),\ -0.213 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.572950 + 0.712022i\)
\(L(\frac12)\) \(\approx\) \(0.572950 + 0.712022i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.31 + 2.37i)T \)
good2 \( 1 + (0.476 - 1.46i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.476 - 1.46i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.190 + 0.138i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.30 - 4.02i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.83 + 5.65i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.73 + 2.71i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 7.49T + 23T^{2} \)
29 \( 1 + (1.54 + 1.12i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.736 - 2.26i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.04 + 3.66i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.51 - 3.28i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + (0.771 - 0.560i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.79 - 8.59i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.83 - 4.96i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.33 - 4.11i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 3.85T + 67T^{2} \)
71 \( 1 + (2.38 + 7.33i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.23 - 3.80i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.163 - 0.502i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.36 - 4.18i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + (3.95 - 12.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.35000582949412150370872480602, −13.61405720342620725596840349226, −11.83990863810097002437522535741, −11.17609648175847544469955304582, −9.484850444696367987562301613656, −8.712689065126223065505905651229, −7.03177062674679116946744546239, −6.76185038977254281236464950227, −5.14267665302715992209516976583, −2.96592272574710577737959721356, 1.54587090049141863590163174685, 3.41797442981747160207817098763, 5.23476786268654704783853889482, 6.77053348669658884742095531004, 8.456444948874643382791324184426, 9.525824805484387533438308794485, 10.35268894495001875172298569809, 11.46219139483417754566890553636, 12.55863503223967517868882497609, 13.02172898955578458384305736159

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