↑

Properties

Label 2-990-11.4-c1-0-0
Degree 22
Conductor 990990
Sign 0.151−0.988i0.151 - 0.988i
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 + (−1.38 − 4.27i)7-s + (0.309 − 0.951i)8-s + 10-s + (−3.28 + 0.458i)11-s + (1.88 + 1.37i)13-s + (−1.38 + 4.27i)14-s + (−0.809 + 0.587i)16-s + (−4.17 + 3.03i)17-s + (−1.85 + 5.71i)19-s + (−0.809 − 0.587i)20-s + (2.92 + 1.55i)22-s + 8.88·23-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (−0.524 − 1.61i)7-s + (0.109 − 0.336i)8-s + 0.316·10-s + (−0.990 + 0.138i)11-s + (0.523 + 0.380i)13-s + (−0.370 + 1.14i)14-s + (−0.202 + 0.146i)16-s + (−1.01 + 0.735i)17-s + (−0.426 + 1.31i)19-s + (−0.180 − 0.131i)20-s + (0.624 + 0.332i)22-s + 1.85·23-s + ⋯

Functional equation

Λ(s)=(990s/2 ΓC(s) L(s)=((0.151−0.988i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(990s/2 ΓC(s+1/2) L(s)=((0.151−0.988i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 990990    =    2⋅32⋅5⋅112 \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.151−0.988i0.151 - 0.988i
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), 0.151−0.988i)(2,\ 990,\ (\ :1/2),\ 0.151 - 0.988i)

Particular Values

L(1)L(1) ≈\approx 0.353597+0.303671i0.353597 + 0.303671i
L(12)L(\frac12) ≈\approx 0.353597+0.303671i0.353597 + 0.303671i
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+(3.28−0.458i)T 1 + (3.28 - 0.458i)T
good7 1+(1.38+4.27i)T+(−5.66+4.11i)T2 1 + (1.38 + 4.27i)T + (-5.66 + 4.11i)T^{2}
13 1+(−1.88−1.37i)T+(4.01+12.3i)T2 1 + (-1.88 - 1.37i)T + (4.01 + 12.3i)T^{2}
17 1+(4.17−3.03i)T+(5.25−16.1i)T2 1 + (4.17 - 3.03i)T + (5.25 - 16.1i)T^{2}
19 1+(1.85−5.71i)T+(−15.3−11.1i)T2 1 + (1.85 - 5.71i)T + (-15.3 - 11.1i)T^{2}
23 1−8.88T+23T2 1 - 8.88T + 23T^{2}
29 1+(2.34+7.21i)T+(−23.4+17.0i)T2 1 + (2.34 + 7.21i)T + (-23.4 + 17.0i)T^{2}
31 1+(−0.897−0.651i)T+(9.57+29.4i)T2 1 + (-0.897 - 0.651i)T + (9.57 + 29.4i)T^{2}
37 1+(−2.45−7.54i)T+(−29.9+21.7i)T2 1 + (-2.45 - 7.54i)T + (-29.9 + 21.7i)T^{2}
41 1+(2.95−9.10i)T+(−33.1−24.0i)T2 1 + (2.95 - 9.10i)T + (-33.1 - 24.0i)T^{2}
43 1+0.0785T+43T2 1 + 0.0785T + 43T^{2}
47 1+(0.896−2.76i)T+(−38.0−27.6i)T2 1 + (0.896 - 2.76i)T + (-38.0 - 27.6i)T^{2}
53 1+(−0.602−0.437i)T+(16.3+50.4i)T2 1 + (-0.602 - 0.437i)T + (16.3 + 50.4i)T^{2}
59 1+(−1.64−5.06i)T+(−47.7+34.6i)T2 1 + (-1.64 - 5.06i)T + (-47.7 + 34.6i)T^{2}
61 1+(12.5−9.08i)T+(18.8−58.0i)T2 1 + (12.5 - 9.08i)T + (18.8 - 58.0i)T^{2}
67 1+1.67T+67T2 1 + 1.67T + 67T^{2}
71 1+(−7.72+5.61i)T+(21.9−67.5i)T2 1 + (-7.72 + 5.61i)T + (21.9 - 67.5i)T^{2}
73 1+(−3.38−10.4i)T+(−59.0+42.9i)T2 1 + (-3.38 - 10.4i)T + (-59.0 + 42.9i)T^{2}
79 1+(−4.42−3.21i)T+(24.4+75.1i)T2 1 + (-4.42 - 3.21i)T + (24.4 + 75.1i)T^{2}
83 1+(6.47−4.70i)T+(25.6−78.9i)T2 1 + (6.47 - 4.70i)T + (25.6 - 78.9i)T^{2}
89 1−7.41T+89T2 1 - 7.41T + 89T^{2}
97 1+(4.64+3.37i)T+(29.9+92.2i)T2 1 + (4.64 + 3.37i)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

−10.28840346845504611300372160230, −9.568576961660428673764455471810, −8.395299372906137965665764024342, −7.79114418331688147440454057795, −6.93301672151967139268956120306, −6.22960014362624463307190883006, −4.56079409675200606682196288586, −3.84663080501607018605792756614, −2.83287652170699657269273888885, −1.27924752292206038614556109869, 0.27382336516234594451699112795, 2.29042448309726141844208487387, 3.14990111290861543625151186806, 4.95513074151439573758436231695, 5.40185382953411459858825010442, 6.52643714624003449604789191338, 7.26146177854550866726726496748, 8.372455242414219262953993115379, 9.060356575173754107492575216650, 9.267858814954989440661181077946

Graph of the ZZ-function along the critical line