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Properties

Label 2-2268-252.167-c0-0-8
Degree 22
Conductor 22682268
Sign 0.0871−0.996i0.0871 - 0.996i
Analytic cond. 1.131871.13187
Root an. cond. 1.063891.06389
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.258 + 0.448i)11-s + (0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (−0.366 + 0.366i)22-s + (−0.707 + 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (1.22 − 0.707i)29-s + (0.965 + 0.258i)32-s + 1.73·37-s + (0.866 − 0.5i)43-s + (−0.448 − 0.258i)44-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.258 + 0.448i)11-s + (0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (−0.366 + 0.366i)22-s + (−0.707 + 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (1.22 − 0.707i)29-s + (0.965 + 0.258i)32-s + 1.73·37-s + (0.866 − 0.5i)43-s + (−0.448 − 0.258i)44-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=((0.0871−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s) L(s)=((0.0871−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: 0.0871−0.996i0.0871 - 0.996i
Analytic conductor: 1.131871.13187
Root analytic conductor: 1.063891.06389
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2268(1511,⋅)\chi_{2268} (1511, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2268, ( :0), 0.0871−0.996i)(2,\ 2268,\ (\ :0),\ 0.0871 - 0.996i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3580314281.358031428
L(12)L(\frac12) ≈\approx 1.3580314281.358031428
L(1)L(1) 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+(−0.258−0.965i)T 1 + (-0.258 - 0.965i)T
3 1 1
7 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
good5 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
11 1+(−0.258−0.448i)T+(−0.5+0.866i)T2 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2}
13 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+T2 1 + T^{2}
19 1+T2 1 + T^{2}
23 1+(0.707−1.22i)T+(−0.5−0.866i)T2 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2}
29 1+(−1.22+0.707i)T+(0.5−0.866i)T2 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1−1.73T+T2 1 - 1.73T + T^{2}
41 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
43 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1−1.93iT−T2 1 - 1.93iT - T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
71 1+1.93T+T2 1 + 1.93T + T^{2}
73 1−T2 1 - T^{2}
79 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{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

−9.233206853699239033630202267032, −8.446305655237215021886722305081, −7.57378401132469505552791869221, −7.33485432899765466560848176764, −6.21351656936421063402118767998, −5.55448875738906137436331039117, −4.52931081138681978805559026796, −4.13278411068795366262171829759, −2.88613039795436815491106179330, −1.32245215164252951105350974779, 1.07282001985636962221338768360, 2.28351530978771990606124622162, 3.01440143136858838978983986230, 4.30927130118260779264627462589, 4.71486802794343878513199806105, 5.79243373364524186894161891716, 6.40857204261009688343535081379, 7.75672708094817211412654537689, 8.586674576146837852350636662391, 8.881958754868903641289227612439

Graph of the ZZ-function along the critical line