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Properties

Label 2-99-9.4-c1-0-4
Degree $2$
Conductor $99$
Sign $1$
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  + (−0.173 − 0.300i)2-s + (1.11 + 1.32i)3-s + (0.939 − 1.62i)4-s + (−0.326 + 0.565i)5-s + (0.205 − 0.565i)6-s + (−0.266 − 0.460i)7-s − 1.34·8-s + (−0.520 + 2.95i)9-s + 0.226·10-s + (0.5 + 0.866i)11-s + (3.20 − 0.565i)12-s + (1.15 − 1.99i)13-s + (−0.0923 + 0.160i)14-s + (−1.11 + 0.196i)15-s + (−1.64 − 2.84i)16-s − 4.41·17-s + ⋯
L(s)  = 1  + (−0.122 − 0.212i)2-s + (0.642 + 0.766i)3-s + (0.469 − 0.813i)4-s + (−0.145 + 0.252i)5-s + (0.0839 − 0.230i)6-s + (−0.100 − 0.174i)7-s − 0.476·8-s + (−0.173 + 0.984i)9-s + 0.0716·10-s + (0.150 + 0.261i)11-s + (0.925 − 0.163i)12-s + (0.319 − 0.553i)13-s + (−0.0246 + 0.0427i)14-s + (−0.287 + 0.0506i)15-s + (−0.411 − 0.712i)16-s − 1.06·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15669\)
\(L(\frac12)\) \(\approx\) \(1.15669\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 - 1.32i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.173 + 0.300i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.326 - 0.565i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.266 + 0.460i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-1.15 + 1.99i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.41T + 17T^{2} \)
19 \( 1 + 4.18T + 19T^{2} \)
23 \( 1 + (0.705 - 1.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.17 + 5.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.08 + 1.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.16T + 37T^{2} \)
41 \( 1 + (-4.65 + 8.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.14 - 10.6i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.93 - 3.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + (-1.61 + 2.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.26 - 7.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.48 - 4.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.98T + 71T^{2} \)
73 \( 1 + 7.49T + 73T^{2} \)
79 \( 1 + (-2.43 - 4.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.29 - 3.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 + (6.32 + 10.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.11233409881310068474863605086, −13.01804083516216763878517296488, −11.33811641512491537451562548090, −10.66427037100759327767608180060, −9.713412905019593754802042315726, −8.676951554736458753380725547181, −7.17809891426627090209318555779, −5.72053030813267742374670059799, −4.13069803706824618099025629107, −2.42509581480708884265128443841, 2.37101318364925647059534818824, 3.95841629004373223610675564355, 6.32538323149485518983050714305, 7.16576415121361910830222974071, 8.498515348238877535176715600957, 8.936597985983079977988058427238, 10.93055533624928347816825680049, 12.09049015043492013796127876828, 12.77877194568126969366914715362, 13.77745100959648994204920313693

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