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Properties

Label 2-1127-161.114-c0-0-8
Degree 22
Conductor 11271127
Sign −0.266+0.963i-0.266 + 0.963i
Analytic cond. 0.5624460.562446
Root an. cond. 0.7499640.749964
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 + (−0.5 − 0.866i)3-s − 0.999·6-s + 8-s + 13-s + (0.5 − 0.866i)16-s + (−0.5 + 0.866i)23-s + (−0.500 − 0.866i)24-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s − 27-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)39-s + 41-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s − 0.999·6-s + 8-s + 13-s + (0.5 − 0.866i)16-s + (−0.5 + 0.866i)23-s + (−0.500 − 0.866i)24-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s − 27-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)39-s + 41-s + ⋯

Functional equation

Λ(s)=(1127s/2 ΓC(s) L(s)=((−0.266+0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1127s/2 ΓC(s) L(s)=((−0.266+0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11271127    =    72⋅237^{2} \cdot 23
Sign: −0.266+0.963i-0.266 + 0.963i
Analytic conductor: 0.5624460.562446
Root analytic conductor: 0.7499640.749964
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1127(275,⋅)\chi_{1127} (275, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1127, ( :0), −0.266+0.963i)(2,\ 1127,\ (\ :0),\ -0.266 + 0.963i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3120891401.312089140
L(12)L(\frac12) ≈\approx 1.3120891401.312089140
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)
bad7 1 1
23 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good2 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
3 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1−T+T2 1 - T + T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
29 1+T+T2 1 + T + 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+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1−T+T2 1 - T + T^{2}
43 1−T2 1 - T^{2}
47 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-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+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1−T2 1 - 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

−10.02929629718358285413206972134, −9.066800266106310562839259573551, −7.85323380701671271106028131465, −7.39183160453245654258911734337, −6.28590637520615007201096102420, −5.64327155241069703588362568038, −4.25375539432778622411418137841, −3.57607872202454958612393336409, −2.26772748487487028694225720026, −1.28408681354923778933539869819, 1.79034456063027282311271813016, 3.63667066355729846682429716603, 4.38397673477583899584194832421, 5.31880029653584820588693884326, 5.83410849858096410964830666373, 6.75089436009230286956591724389, 7.61004858035604702656352035655, 8.523310950768372098375891448168, 9.564682515641307364691732435632, 10.34341293884542141807261731112

Graph of the ZZ-function along the critical line