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Properties

Label 2-99-99.32-c1-0-2
Degree $2$
Conductor $99$
Sign $0.0662 - 0.997i$
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  + (0.618 + 1.07i)2-s + (−1.04 + 1.37i)3-s + (0.234 − 0.406i)4-s + (1.78 + 1.02i)5-s + (−2.12 − 0.271i)6-s + (−3.59 + 2.07i)7-s + 3.05·8-s + (−0.800 − 2.89i)9-s + 2.54i·10-s + (−0.0426 − 3.31i)11-s + (0.314 + 0.749i)12-s + (0.747 + 0.431i)13-s + (−4.44 − 2.56i)14-s + (−3.28 + 1.37i)15-s + (1.42 + 2.46i)16-s + 4.25·17-s + ⋯
L(s)  = 1  + (0.437 + 0.757i)2-s + (−0.605 + 0.795i)3-s + (0.117 − 0.203i)4-s + (0.797 + 0.460i)5-s + (−0.867 − 0.110i)6-s + (−1.35 + 0.784i)7-s + 1.08·8-s + (−0.266 − 0.963i)9-s + 0.805i·10-s + (−0.0128 − 0.999i)11-s + (0.0906 + 0.216i)12-s + (0.207 + 0.119i)13-s + (−1.18 − 0.685i)14-s + (−0.849 + 0.355i)15-s + (0.355 + 0.615i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0662 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0662 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.0662 - 0.997i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.0662 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.834923 + 0.781359i\)
\(L(\frac12)\) \(\approx\) \(0.834923 + 0.781359i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.04 - 1.37i)T \)
11 \( 1 + (0.0426 + 3.31i)T \)
good2 \( 1 + (-0.618 - 1.07i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.78 - 1.02i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.59 - 2.07i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (-0.747 - 0.431i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.25T + 17T^{2} \)
19 \( 1 + 5.55iT - 19T^{2} \)
23 \( 1 + (2.25 + 1.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.44 - 2.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.40 - 4.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 + (2.82 - 4.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.96 - 1.13i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.44 - 0.831i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.28iT - 53T^{2} \)
59 \( 1 + (-5.79 - 3.34i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.78 - 3.34i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.08 - 1.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.36iT - 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 + (-9.22 + 5.32i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.10 - 5.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.44iT - 89T^{2} \)
97 \( 1 + (-5.27 - 9.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.29852027022529808751784553090, −13.41242795024601881031596993645, −12.07436067241311761135782242706, −10.72311996590837542480268522903, −9.991228357108453849534015673611, −8.906418465268446371805972689971, −6.70698669129051054971736058646, −6.07455212397173788467517052801, −5.22718324912552406595290800737, −3.21542584278489075882551045571, 1.78699016243529828360183974755, 3.67126016196729003032845929805, 5.46057077375382649127951400921, 6.76881073986111059779459992755, 7.80144789991177750931867814234, 9.841880899355802498572286636124, 10.43472166264392208653734247623, 11.99191868317206055481404765569, 12.57049665885098108945947255353, 13.30416032622046653264942154712

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