↑

Properties

Label 2-99-11.3-c3-0-8
Degree $2$
Conductor $99$
Sign $0.422 + 0.906i$
Analytic cond. $5.84118$
Root an. cond. $2.41685$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.523 + 0.380i)2-s + (−2.34 + 7.21i)4-s + (−9.01 − 6.54i)5-s + (8.07 − 24.8i)7-s + (−3.11 − 9.58i)8-s + 7.20·10-s + (36.0 − 5.31i)11-s + (43.2 − 31.4i)13-s + (5.22 + 16.0i)14-s + (−43.7 − 31.8i)16-s + (18.9 + 13.7i)17-s + (−21.8 − 67.3i)19-s + (68.3 − 49.6i)20-s + (−16.8 + 16.5i)22-s − 164.·23-s + ⋯
L(s)  = 1  + (−0.185 + 0.134i)2-s + (−0.292 + 0.901i)4-s + (−0.806 − 0.585i)5-s + (0.436 − 1.34i)7-s + (−0.137 − 0.423i)8-s + 0.227·10-s + (0.989 − 0.145i)11-s + (0.923 − 0.671i)13-s + (0.0997 + 0.307i)14-s + (−0.684 − 0.497i)16-s + (0.270 + 0.196i)17-s + (−0.264 − 0.813i)19-s + (0.764 − 0.555i)20-s + (−0.163 + 0.159i)22-s − 1.48·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.422 + 0.906i$
Analytic conductor: \(5.84118\)
Root analytic conductor: \(2.41685\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :3/2),\ 0.422 + 0.906i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.891548 - 0.567822i\)
\(L(\frac12)\) \(\approx\) \(0.891548 - 0.567822i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-36.0 + 5.31i)T \)
good2 \( 1 + (0.523 - 0.380i)T + (2.47 - 7.60i)T^{2} \)
5 \( 1 + (9.01 + 6.54i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (-8.07 + 24.8i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (-43.2 + 31.4i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-18.9 - 13.7i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (21.8 + 67.3i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + 164.T + 1.21e4T^{2} \)
29 \( 1 + (-67.5 + 207. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (-62.0 + 45.0i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (87.5 - 269. i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (1.50 + 4.61i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + 333.T + 7.95e4T^{2} \)
47 \( 1 + (-121. - 374. i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-123. + 89.3i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-237. + 729. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (-287. - 209. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 - 102.T + 3.00e5T^{2} \)
71 \( 1 + (-504. - 366. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (95.7 - 294. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (-517. + 375. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (-233. - 169. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + 184.T + 7.04e5T^{2} \)
97 \( 1 + (515. - 374. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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

−13.28724106722898979139027883926, −12.10151423936794699998314980753, −11.30349495104718837014735216023, −9.900411016062100504856337676633, −8.368351166543914633221355405535, −7.970116669740604855620791670873, −6.60867442415890543126393879583, −4.40684819299942353704182234403, −3.72079209624106745531914731642, −0.69565277785851343745301982854, 1.78762179537612582089628097815, 3.88800350485130563501896733421, 5.52186835758060432085235190682, 6.63192604595208001595682756831, 8.353457869854073644136944264129, 9.153814683935868282834850639294, 10.45672313848952541301357190913, 11.56604369330660049035750173033, 12.11963172960743263817263301555, 13.96730565969226906137686398167

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