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Properties

Label 2-99-33.2-c1-0-0
Degree $2$
Conductor $99$
Sign $-0.999 - 0.00310i$
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  + (−0.726 + 2.23i)2-s + (−2.85 − 2.07i)4-s + (−2.13 + 0.694i)5-s + (−2.38 + 3.27i)7-s + (2.90 − 2.10i)8-s − 5.28i·10-s + (3.31 − 0.0200i)11-s + (4.42 + 1.43i)13-s + (−5.59 − 7.70i)14-s + (0.427 + 1.31i)16-s + (0.0235 + 0.0725i)17-s + (1.40 + 1.93i)19-s + (7.53 + 2.44i)20-s + (−2.36 + 7.42i)22-s + 3.22i·23-s + ⋯
L(s)  = 1  + (−0.513 + 1.58i)2-s + (−1.42 − 1.03i)4-s + (−0.956 + 0.310i)5-s + (−0.899 + 1.23i)7-s + (1.02 − 0.745i)8-s − 1.67i·10-s + (0.999 − 0.00604i)11-s + (1.22 + 0.398i)13-s + (−1.49 − 2.05i)14-s + (0.106 + 0.328i)16-s + (0.00571 + 0.0175i)17-s + (0.323 + 0.444i)19-s + (1.68 + 0.547i)20-s + (−0.504 + 1.58i)22-s + 0.672i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $-0.999 - 0.00310i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ -0.999 - 0.00310i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000880781 + 0.568205i\)
\(L(\frac12)\) \(\approx\) \(0.000880781 + 0.568205i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 + 0.0200i)T \)
good2 \( 1 + (0.726 - 2.23i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (2.13 - 0.694i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.38 - 3.27i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-4.42 - 1.43i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.0235 - 0.0725i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.40 - 1.93i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.22iT - 23T^{2} \)
29 \( 1 + (1.48 + 1.08i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.517 + 1.59i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.87 + 4.27i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.82 + 4.96i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.28iT - 43T^{2} \)
47 \( 1 + (-3.65 - 5.02i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.16 + 0.379i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.341 + 0.469i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.59 - 1.16i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + (1.06 - 0.346i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.82 + 10.7i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.627 - 0.203i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.15 - 9.71i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 6.58iT - 89T^{2} \)
97 \( 1 + (-5.08 + 15.6i)T + (-78.4 - 57.0i)T^{2} \)
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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

−14.82329736041423648954689826432, −13.82385061502975220616048624790, −12.33090517676581196959596342690, −11.31763942722825184534517235082, −9.463378423603972416847587503809, −8.838386943022131644960127744666, −7.69750669254702180655329149614, −6.53124109005508026100759207173, −5.73717545520714436180078245035, −3.73667479960224348845956228086, 0.855652997329293444855679185769, 3.43283514169905169840176564669, 4.12774478154439322216338921883, 6.72540443750132085188246097818, 8.232672145853905120298687386574, 9.274735293173336602986796233262, 10.37661599593391814969964488364, 11.18008677430688746017387332576, 12.11986899844358967670279363634, 13.03546170021631902199781450422

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