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Properties

Label 2-2268-21.17-c1-0-12
Degree 22
Conductor 22682268
Sign 0.998+0.0484i0.998 + 0.0484i
Analytic cond. 18.110018.1100
Root an. cond. 4.255594.25559
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  + (0.349 + 0.605i)5-s + (−2.02 − 1.70i)7-s + (−0.229 − 0.132i)11-s + 1.31i·13-s + (−1.86 + 3.22i)17-s + (−0.382 + 0.220i)19-s + (4.29 − 2.48i)23-s + (2.25 − 3.90i)25-s + 0.315i·29-s + (4.85 + 2.80i)31-s + (0.322 − 1.82i)35-s + (−0.351 − 0.608i)37-s + 10.7·41-s − 7.46·43-s + (3.50 + 6.06i)47-s + ⋯
L(s)  = 1  + (0.156 + 0.270i)5-s + (−0.765 − 0.643i)7-s + (−0.0692 − 0.0399i)11-s + 0.364i·13-s + (−0.452 + 0.783i)17-s + (−0.0877 + 0.0506i)19-s + (0.896 − 0.517i)23-s + (0.451 − 0.781i)25-s + 0.0585i·29-s + (0.872 + 0.503i)31-s + (0.0545 − 0.308i)35-s + (−0.0577 − 0.0999i)37-s + 1.68·41-s − 1.13·43-s + (0.510 + 0.884i)47-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=((0.998+0.0484i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0484i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s+1/2) L(s)=((0.998+0.0484i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0484i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: 0.998+0.0484i0.998 + 0.0484i
Analytic conductor: 18.110018.1100
Root analytic conductor: 4.255594.25559
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2268(1781,⋅)\chi_{2268} (1781, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2268, ( :1/2), 0.998+0.0484i)(2,\ 2268,\ (\ :1/2),\ 0.998 + 0.0484i)

Particular Values

L(1)L(1) ≈\approx 1.5907573171.590757317
L(12)L(\frac12) ≈\approx 1.5907573171.590757317
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 1
3 1 1
7 1+(2.02+1.70i)T 1 + (2.02 + 1.70i)T
good5 1+(−0.349−0.605i)T+(−2.5+4.33i)T2 1 + (-0.349 - 0.605i)T + (-2.5 + 4.33i)T^{2}
11 1+(0.229+0.132i)T+(5.5+9.52i)T2 1 + (0.229 + 0.132i)T + (5.5 + 9.52i)T^{2}
13 1−1.31iT−13T2 1 - 1.31iT - 13T^{2}
17 1+(1.86−3.22i)T+(−8.5−14.7i)T2 1 + (1.86 - 3.22i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.382−0.220i)T+(9.5−16.4i)T2 1 + (0.382 - 0.220i)T + (9.5 - 16.4i)T^{2}
23 1+(−4.29+2.48i)T+(11.5−19.9i)T2 1 + (-4.29 + 2.48i)T + (11.5 - 19.9i)T^{2}
29 1−0.315iT−29T2 1 - 0.315iT - 29T^{2}
31 1+(−4.85−2.80i)T+(15.5+26.8i)T2 1 + (-4.85 - 2.80i)T + (15.5 + 26.8i)T^{2}
37 1+(0.351+0.608i)T+(−18.5+32.0i)T2 1 + (0.351 + 0.608i)T + (-18.5 + 32.0i)T^{2}
41 1−10.7T+41T2 1 - 10.7T + 41T^{2}
43 1+7.46T+43T2 1 + 7.46T + 43T^{2}
47 1+(−3.50−6.06i)T+(−23.5+40.7i)T2 1 + (-3.50 - 6.06i)T + (-23.5 + 40.7i)T^{2}
53 1+(−8.51−4.91i)T+(26.5+45.8i)T2 1 + (-8.51 - 4.91i)T + (26.5 + 45.8i)T^{2}
59 1+(−6.73+11.6i)T+(−29.5−51.0i)T2 1 + (-6.73 + 11.6i)T + (-29.5 - 51.0i)T^{2}
61 1+(−4.89+2.82i)T+(30.5−52.8i)T2 1 + (-4.89 + 2.82i)T + (30.5 - 52.8i)T^{2}
67 1+(−2.97+5.14i)T+(−33.5−58.0i)T2 1 + (-2.97 + 5.14i)T + (-33.5 - 58.0i)T^{2}
71 1+13.4iT−71T2 1 + 13.4iT - 71T^{2}
73 1+(6.66+3.84i)T+(36.5+63.2i)T2 1 + (6.66 + 3.84i)T + (36.5 + 63.2i)T^{2}
79 1+(0.698+1.20i)T+(−39.5+68.4i)T2 1 + (0.698 + 1.20i)T + (-39.5 + 68.4i)T^{2}
83 1−7.44T+83T2 1 - 7.44T + 83T^{2}
89 1+(−5.59−9.68i)T+(−44.5+77.0i)T2 1 + (-5.59 - 9.68i)T + (-44.5 + 77.0i)T^{2}
97 1−10.6iT−97T2 1 - 10.6iT - 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

−9.051021726886360543186195113999, −8.288641366588664939630321054603, −7.37154538617951372630236820724, −6.55274254786892261931145159787, −6.20560772087462035715810616461, −4.92397024675129136597237183521, −4.13705762019720857234454408936, −3.21784828992933651377594807705, −2.26690182003494364953443260648, −0.804081141609464637527659313183, 0.825836180540485080648224827087, 2.35550176469284964164408431567, 3.08484041909640907961767776983, 4.17472483970067175039282940091, 5.24719717042522998829111767678, 5.73759380190345488117357473812, 6.79728588961705384336421567266, 7.32476667249430187959993769338, 8.504876653930116417462774799159, 8.973193929166232333673410938871

Graph of the ZZ-function along the critical line