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Properties

Label 2-2268-252.83-c0-0-8
Degree 22
Conductor 22682268
Sign 0.342+0.939i0.342 + 0.939i
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 2·17-s + 19-s + (0.499 − 0.866i)20-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s − 0.999·28-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s − 0.999·8-s + 0.999·10-s + (−0.5 + 0.866i)11-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 2·17-s + 19-s + (0.499 − 0.866i)20-s + (0.499 + 0.866i)22-s + (−0.5 − 0.866i)23-s − 0.999·28-s + ⋯

Functional equation

Λ(s)=(2268s/2 ΓC(s) L(s)=((0.342+0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2268s/2 ΓC(s) L(s)=((0.342+0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22682268    =    22⋅34⋅72^{2} \cdot 3^{4} \cdot 7
Sign: 0.342+0.939i0.342 + 0.939i
Analytic conductor: 1.131871.13187
Root analytic conductor: 1.063891.06389
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2268(755,⋅)\chi_{2268} (755, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2268, ( :0), 0.342+0.939i)(2,\ 2268,\ (\ :0),\ 0.342 + 0.939i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6718132201.671813220
L(12)L(\frac12) ≈\approx 1.6718132201.671813220
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1 1
7 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good5 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
11 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
13 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1−2T+T2 1 - 2T + T^{2}
19 1−T+T2 1 - T + T^{2}
23 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
29 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
31 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
37 1+T+T2 1 + T + T^{2}
41 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1−T2 1 - 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+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−T+T2 1 - T + T^{2}
73 1−T2 1 - T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1+T+T2 1 + T + 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

−9.633467101347565511154702836330, −8.196326751296965268077681621857, −7.49958351367229377470887375163, −6.67632847316378803248875895481, −5.68826070690439759320572913694, −5.02979612245365128530448139990, −4.06108478928931631392451988095, −3.20879248805190118779231409645, −2.32213462980658126936753311184, −1.23122936992700661333303101077, 1.39950496191664247000278156541, 2.92511609828972627507093950251, 3.69450541689916265986934381579, 5.07041090306899905344249599317, 5.51453117300928735154655764194, 5.74277004547041211131294746063, 7.09217965295457934381414450408, 7.925445301327123315978242036883, 8.386964245086595000936451757591, 9.206498672716002321955975821388

Graph of the ZZ-function along the critical line