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Properties

Label 2-990-33.32-c1-0-7
Degree 22
Conductor 990990
Sign 0.978+0.207i0.978 + 0.207i
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + i·5-s − 1.67i·7-s − 8-s − i·10-s + (2.25 + 2.43i)11-s − 2.95i·13-s + 1.67i·14-s + 16-s + 0.828·17-s − 2.26i·19-s + i·20-s + (−2.25 − 2.43i)22-s + 0.720i·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.447i·5-s − 0.632i·7-s − 0.353·8-s − 0.316i·10-s + (0.678 + 0.734i)11-s − 0.819i·13-s + 0.447i·14-s + 0.250·16-s + 0.200·17-s − 0.518i·19-s + 0.223i·20-s + (−0.480 − 0.519i)22-s + 0.150i·23-s + ⋯

Functional equation

Λ(s)=(990s/2 ΓC(s) L(s)=((0.978+0.207i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(990s/2 ΓC(s+1/2) L(s)=((0.978+0.207i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 990990    =    2⋅32⋅5⋅112 \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.978+0.207i0.978 + 0.207i
Analytic conductor: 7.905187.90518
Root analytic conductor: 2.811612.81161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ990(791,⋅)\chi_{990} (791, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 990, ( :1/2), 0.978+0.207i)(2,\ 990,\ (\ :1/2),\ 0.978 + 0.207i)

Particular Values

L(1)L(1) ≈\approx 1.17777−0.123619i1.17777 - 0.123619i
L(12)L(\frac12) ≈\approx 1.17777−0.123619i1.17777 - 0.123619i
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+T 1 + T
3 1 1
5 1−iT 1 - iT
11 1+(−2.25−2.43i)T 1 + (-2.25 - 2.43i)T
good7 1+1.67iT−7T2 1 + 1.67iT - 7T^{2}
13 1+2.95iT−13T2 1 + 2.95iT - 13T^{2}
17 1−0.828T+17T2 1 - 0.828T + 17T^{2}
19 1+2.26iT−19T2 1 + 2.26iT - 19T^{2}
23 1−0.720iT−23T2 1 - 0.720iT - 23T^{2}
29 1−2.36T+29T2 1 - 2.36T + 29T^{2}
31 1−4.36T+31T2 1 - 4.36T + 31T^{2}
37 1−4.69T+37T2 1 - 4.69T + 37T^{2}
41 1−1.21T+41T2 1 - 1.21T + 41T^{2}
43 1+5.08iT−43T2 1 + 5.08iT - 43T^{2}
47 1+0.911iT−47T2 1 + 0.911iT - 47T^{2}
53 1+3.71iT−53T2 1 + 3.71iT - 53T^{2}
59 1−2.69iT−59T2 1 - 2.69iT - 59T^{2}
61 1−8.74iT−61T2 1 - 8.74iT - 61T^{2}
67 1−9.52T+67T2 1 - 9.52T + 67T^{2}
71 1−4.93iT−71T2 1 - 4.93iT - 71T^{2}
73 1+6.36iT−73T2 1 + 6.36iT - 73T^{2}
79 1−2.46iT−79T2 1 - 2.46iT - 79T^{2}
83 1−8.85T+83T2 1 - 8.85T + 83T^{2}
89 1+7.84iT−89T2 1 + 7.84iT - 89T^{2}
97 1−17.5T+97T2 1 - 17.5T + 97T^{2}
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   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

−10.05479041889350996619595711669, −9.222337474597384034115657826187, −8.269855050199098254260972096690, −7.41427897254376522815932643601, −6.83705107803337846146619308688, −5.88268787245037051463225418828, −4.62092853677126427150565252560, −3.52076391008479936664220320579, −2.36520577366223383751054961149, −0.902693357665715264627831594843, 1.06783812362714290096090078442, 2.34362438418320770562685414790, 3.62801298228091702881509133188, 4.79534150361546660449232842774, 5.98163914456072388839402144006, 6.53627194577047402017788060981, 7.75544328902837023644801193553, 8.485787058703879294404583929847, 9.156371169053805923228651042827, 9.768106991128600612816903441741

Graph of the ZZ-function along the critical line