↑

Properties

Label 2-99-9.4-c1-0-0
Degree $2$
Conductor $99$
Sign $0.545 - 0.838i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 − 1.86i)2-s + (−0.635 + 1.61i)3-s + (−1.32 + 2.29i)4-s + (−1.81 + 3.13i)5-s + (3.69 − 0.550i)6-s + (1.13 + 1.96i)7-s + 1.39·8-s + (−2.19 − 2.04i)9-s + 7.81·10-s + (−0.5 − 0.866i)11-s + (−2.85 − 3.58i)12-s + (−0.619 + 1.07i)13-s + (2.44 − 4.23i)14-s + (−3.90 − 4.91i)15-s + (1.14 + 1.98i)16-s + 5.69·17-s + ⋯
L(s)  = 1  + (−0.762 − 1.32i)2-s + (−0.366 + 0.930i)3-s + (−0.661 + 1.14i)4-s + (−0.810 + 1.40i)5-s + (1.50 − 0.224i)6-s + (0.429 + 0.743i)7-s + 0.492·8-s + (−0.730 − 0.682i)9-s + 2.47·10-s + (−0.150 − 0.261i)11-s + (−0.823 − 1.03i)12-s + (−0.171 + 0.297i)13-s + (0.654 − 1.13i)14-s + (−1.00 − 1.26i)15-s + (0.286 + 0.495i)16-s + 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.545 - 0.838i$
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: \((2,\ 99,\ (\ :1/2),\ 0.545 - 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.406643 + 0.220626i\)
\(L(\frac12)\) \(\approx\) \(0.406643 + 0.220626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.635 - 1.61i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.07 + 1.86i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.81 - 3.13i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.13 - 1.96i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (0.619 - 1.07i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.69T + 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
23 \( 1 + (2.95 - 5.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.75 - 3.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.25 + 2.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.333T + 37T^{2} \)
41 \( 1 + (-4.98 + 8.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.57 - 6.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.34 - 7.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.16T + 53T^{2} \)
59 \( 1 + (1.45 - 2.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.13 + 5.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.68 - 8.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 - 4.31T + 73T^{2} \)
79 \( 1 + (0.708 + 1.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.37 - 2.38i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.77T + 89T^{2} \)
97 \( 1 + (-1.27 - 2.21i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.33315307249265669838506551771, −12.20828000343087903018011817646, −11.63775572938113064529535992369, −10.84158293943188274932993400117, −10.14598930348002935118959637153, −9.022349019410599760778603775452, −7.76826253068596008981390347522, −5.88593171555966075386645531454, −3.86843007748427001023293042077, −2.77760752557498646334963294429, 0.74853076515329875380635694455, 4.67025176343136594905340310929, 5.89825725737881031304735430165, 7.35431802402363051393858774595, 7.974017023043989234759629850325, 8.702901172203196726415357815155, 10.30483925835444743127085595539, 11.94849774078633707559539573124, 12.58971666681266366980497958140, 13.88820550180425615922470065802

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