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Properties

Label 2-1127-161.160-c1-0-65
Degree 22
Conductor 11271127
Sign −0.156−0.987i-0.156 - 0.987i
Analytic cond. 8.999148.99914
Root an. cond. 2.999852.99985
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  − 2.54·2-s − 3.34i·3-s + 4.49·4-s + 8.53i·6-s − 6.36·8-s − 8.20·9-s − 15.0i·12-s − 6.77i·13-s + 7.23·16-s + 20.9·18-s + 4.79·23-s + 21.3i·24-s − 5·25-s + 17.2i·26-s + 17.4i·27-s + ⋯
L(s)  = 1  − 1.80·2-s − 1.93i·3-s + 2.24·4-s + 3.48i·6-s − 2.25·8-s − 2.73·9-s − 4.34i·12-s − 1.87i·13-s + 1.80·16-s + 4.92·18-s + 1.00·23-s + 4.35i·24-s − 25-s + 3.38i·26-s + 3.34i·27-s + ⋯

Functional equation

Λ(s)=(1127s/2 ΓC(s) L(s)=((−0.156−0.987i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1127s/2 ΓC(s+1/2) L(s)=((−0.156−0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11271127    =    72⋅237^{2} \cdot 23
Sign: −0.156−0.987i-0.156 - 0.987i
Analytic conductor: 8.999148.99914
Root analytic conductor: 2.999852.99985
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1127(1126,⋅)\chi_{1127} (1126, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1127, ( :1/2), −0.156−0.987i)(2,\ 1127,\ (\ :1/2),\ -0.156 - 0.987i)

Particular Values

L(1)L(1) ≈\approx 0.25103348830.2510334883
L(12)L(\frac12) ≈\approx 0.25103348830.2510334883
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)
bad7 1 1
23 1−4.79T 1 - 4.79T
good2 1+2.54T+2T2 1 + 2.54T + 2T^{2}
3 1+3.34iT−3T2 1 + 3.34iT - 3T^{2}
5 1+5T2 1 + 5T^{2}
11 1−11T2 1 - 11T^{2}
13 1+6.77iT−13T2 1 + 6.77iT - 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1+19T2 1 + 19T^{2}
29 1+6.70T+29T2 1 + 6.70T + 29T^{2}
31 1+10.1iT−31T2 1 + 10.1iT - 31T^{2}
37 1−37T2 1 - 37T^{2}
41 1−0.987iT−41T2 1 - 0.987iT - 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1−8.61iT−47T2 1 - 8.61iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1−4.26iT−59T2 1 - 4.26iT - 59T^{2}
61 1+61T2 1 + 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+14.0T+71T2 1 + 14.0T + 71T^{2}
73 1−1.17iT−73T2 1 - 1.17iT - 73T^{2}
79 1−79T2 1 - 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+97T2 1 + 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

−8.990749632574807053057190339079, −8.191206582225484632439265772284, −7.62537264241238050497832295304, −7.30230901421023206290924431347, −6.15470831196934065236023506876, −5.66957570581116654291420145557, −3.06906820930652488472601553333, −2.23488961508807332227161259283, −1.15468526990917564733860842361, −0.21556333019660758790878453055, 1.91991739721287333490679969570, 3.22294624526648296090049032042, 4.24759569664494539459509241107, 5.31377875159097420134897193662, 6.43849482232625393709164669968, 7.34015648628121494422311406722, 8.575275729840034868361088559003, 8.956888360528917366720892846612, 9.536248482732909854396453187064, 10.14195159101239665595191089962

Graph of the ZZ-function along the critical line