Properties

Label 2-99-99.16-c1-0-5
Degree $2$
Conductor $99$
Sign $0.999 + 0.0222i$
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  + (0.373 + 0.0794i)2-s + (1.72 + 0.181i)3-s + (−1.69 − 0.754i)4-s + (1.20 − 0.256i)5-s + (0.629 + 0.204i)6-s + (0.104 + 0.994i)7-s + (−1.19 − 0.865i)8-s + (2.93 + 0.623i)9-s + 0.472·10-s + (−3.01 − 1.38i)11-s + (−2.78 − 1.60i)12-s + (−4.33 + 4.80i)13-s + (−0.0399 + 0.379i)14-s + (2.12 − 0.223i)15-s + (2.10 + 2.33i)16-s + (1.5 − 4.61i)17-s + ⋯
L(s)  = 1  + (0.264 + 0.0561i)2-s + (0.994 + 0.104i)3-s + (−0.846 − 0.377i)4-s + (0.540 − 0.114i)5-s + (0.256 + 0.0834i)6-s + (0.0395 + 0.375i)7-s + (−0.421 − 0.305i)8-s + (0.978 + 0.207i)9-s + 0.149·10-s + (−0.908 − 0.417i)11-s + (−0.802 − 0.463i)12-s + (−1.20 + 1.33i)13-s + (−0.0106 + 0.101i)14-s + (0.549 − 0.0577i)15-s + (0.526 + 0.584i)16-s + (0.363 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.999 + 0.0222i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.999 + 0.0222i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29922 - 0.0144644i\)
\(L(\frac12)\) \(\approx\) \(1.29922 - 0.0144644i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.181i)T \)
11 \( 1 + (3.01 + 1.38i)T \)
good2 \( 1 + (-0.373 - 0.0794i)T + (1.82 + 0.813i)T^{2} \)
5 \( 1 + (-1.20 + 0.256i)T + (4.56 - 2.03i)T^{2} \)
7 \( 1 + (-0.104 - 0.994i)T + (-6.84 + 1.45i)T^{2} \)
13 \( 1 + (4.33 - 4.80i)T + (-1.35 - 12.9i)T^{2} \)
17 \( 1 + (-1.5 + 4.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.30 + 3.99i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.507 - 4.82i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-0.413 + 0.459i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-4.11 + 2.99i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.264 + 2.51i)T + (-40.1 - 8.52i)T^{2} \)
43 \( 1 + (0.927 - 1.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.9 + 4.85i)T + (31.4 - 34.9i)T^{2} \)
53 \( 1 + (-1.26 - 3.88i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.47 - 0.658i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-7.26 - 8.06i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (3 + 5.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.899 - 2.76i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.118 + 0.0857i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (10.3 + 2.19i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (6.53 + 7.25i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + (5.86 + 1.24i)T + (88.6 + 39.4i)T^{2} \)
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

−14.01432334684439153276279893191, −13.19724137394970933369055841612, −12.11621532646198562605935509505, −10.28239897915248852369018601114, −9.434054159550483990935825475823, −8.699788214578980432527569232617, −7.25457108306543526435091046111, −5.49419391252283575947147137615, −4.35635264462464371282617043427, −2.48485563390693407317802539316, 2.63884421669695572618177200637, 4.11193053311449633801127247473, 5.56325286848054776189836955105, 7.61311879159866309710557445783, 8.199818317609371857041896017372, 9.749447626479663857548311397501, 10.20167844088658934032947798674, 12.34810123411433670828992775998, 13.03539481800147395420880231294, 13.78570274734942390164774538510

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