↑

Properties

Label 2-990-11.4-c1-0-10
Degree 22
Conductor 990990
Sign 0.642−0.766i0.642 - 0.766i
Analytic cond. 7.905187.90518
Root an. cond. 2.811612.81161
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−0.309 + 0.951i)8-s + 10-s + (3.04 + 1.31i)11-s + (1.11 + 0.812i)13-s + (−0.809 + 0.587i)16-s + (−1.30 + 0.951i)17-s + (0.618 − 1.90i)19-s + (0.809 + 0.587i)20-s + (1.69 + 2.85i)22-s + 4.85·23-s + (0.309 − 0.951i)25-s + (0.427 + 1.31i)26-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.109 + 0.336i)8-s + 0.316·10-s + (0.918 + 0.396i)11-s + (0.310 + 0.225i)13-s + (−0.202 + 0.146i)16-s + (−0.317 + 0.230i)17-s + (0.141 − 0.436i)19-s + (0.180 + 0.131i)20-s + (0.360 + 0.608i)22-s + 1.01·23-s + (0.0618 − 0.190i)25-s + (0.0837 + 0.257i)26-s + ⋯

Functional equation

Λ(s)=(990s/2 ΓC(s) L(s)=((0.642−0.766i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(990s/2 ΓC(s+1/2) L(s)=((0.642−0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 990990    =    2⋅32⋅5⋅112 \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.642−0.766i0.642 - 0.766i
Analytic conductor: 7.905187.90518
Root analytic conductor: 2.811612.81161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ990(631,⋅)\chi_{990} (631, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 990, ( :1/2), 0.642−0.766i)(2,\ 990,\ (\ :1/2),\ 0.642 - 0.766i)

Particular Values

L(1)L(1) ≈\approx 2.25221+1.05060i2.25221 + 1.05060i
L(12)L(\frac12) ≈\approx 2.25221+1.05060i2.25221 + 1.05060i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
3 1 1
5 1+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
11 1+(−3.04−1.31i)T 1 + (-3.04 - 1.31i)T
good7 1+(−5.66+4.11i)T2 1 + (-5.66 + 4.11i)T^{2}
13 1+(−1.11−0.812i)T+(4.01+12.3i)T2 1 + (-1.11 - 0.812i)T + (4.01 + 12.3i)T^{2}
17 1+(1.30−0.951i)T+(5.25−16.1i)T2 1 + (1.30 - 0.951i)T + (5.25 - 16.1i)T^{2}
19 1+(−0.618+1.90i)T+(−15.3−11.1i)T2 1 + (-0.618 + 1.90i)T + (-15.3 - 11.1i)T^{2}
23 1−4.85T+23T2 1 - 4.85T + 23T^{2}
29 1+(−2.04−6.29i)T+(−23.4+17.0i)T2 1 + (-2.04 - 6.29i)T + (-23.4 + 17.0i)T^{2}
31 1+(−3.35−2.43i)T+(9.57+29.4i)T2 1 + (-3.35 - 2.43i)T + (9.57 + 29.4i)T^{2}
37 1+(2.11+6.51i)T+(−29.9+21.7i)T2 1 + (2.11 + 6.51i)T + (-29.9 + 21.7i)T^{2}
41 1+(1.14−3.52i)T+(−33.1−24.0i)T2 1 + (1.14 - 3.52i)T + (-33.1 - 24.0i)T^{2}
43 1+2.85T+43T2 1 + 2.85T + 43T^{2}
47 1+(0.336−1.03i)T+(−38.0−27.6i)T2 1 + (0.336 - 1.03i)T + (-38.0 - 27.6i)T^{2}
53 1+(−8.85−6.43i)T+(16.3+50.4i)T2 1 + (-8.85 - 6.43i)T + (16.3 + 50.4i)T^{2}
59 1+(−0.354−1.08i)T+(−47.7+34.6i)T2 1 + (-0.354 - 1.08i)T + (-47.7 + 34.6i)T^{2}
61 1+(−6.23+4.53i)T+(18.8−58.0i)T2 1 + (-6.23 + 4.53i)T + (18.8 - 58.0i)T^{2}
67 1+12.6T+67T2 1 + 12.6T + 67T^{2}
71 1+(8.85−6.43i)T+(21.9−67.5i)T2 1 + (8.85 - 6.43i)T + (21.9 - 67.5i)T^{2}
73 1+(−59.0+42.9i)T2 1 + (-59.0 + 42.9i)T^{2}
79 1+(9.78+7.10i)T+(24.4+75.1i)T2 1 + (9.78 + 7.10i)T + (24.4 + 75.1i)T^{2}
83 1+(0.618−0.449i)T+(25.6−78.9i)T2 1 + (0.618 - 0.449i)T + (25.6 - 78.9i)T^{2}
89 1−4.47T+89T2 1 - 4.47T + 89T^{2}
97 1+(7.47+5.42i)T+(29.9+92.2i)T2 1 + (7.47 + 5.42i)T + (29.9 + 92.2i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.08963418148168602374785554811, −8.970254968767420111117263601741, −8.665694681036309870192591120905, −7.25407263594069839836293448993, −6.74893956463561392743672309671, −5.78886987215884889647588878463, −4.86566336007610893638049635905, −4.03929999493829922005524602986, −2.87313905575613649903829905464, −1.45119008401616600152855906216, 1.14665834434289563582481104869, 2.50918357540366020083343978175, 3.51980692524188889470907718947, 4.47004597974096156049519721191, 5.56283835649289482379076930287, 6.33384742649519275826097155349, 7.09858510615976771371242110813, 8.338912518341349291336864531763, 9.163671975780853956982685755944, 10.03199363793961048408334808790

Graph of the ZZ-function along the critical line