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Properties

Label 2-33e2-33.32-c1-0-1
Degree 22
Conductor 10891089
Sign −0.972+0.231i-0.972 + 0.231i
Analytic cond. 8.695708.69570
Root an. cond. 2.948842.94884
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  − 2.43·2-s + 3.93·4-s + 3.79i·5-s + 0.367i·7-s − 4.71·8-s − 9.24i·10-s − 0.948i·13-s − 0.896i·14-s + 3.61·16-s − 3.43·17-s + 4.26i·19-s + 14.9i·20-s + 4.96i·23-s − 9.40·25-s + 2.31i·26-s + ⋯
L(s)  = 1  − 1.72·2-s + 1.96·4-s + 1.69i·5-s + 0.139i·7-s − 1.66·8-s − 2.92i·10-s − 0.263i·13-s − 0.239i·14-s + 0.904·16-s − 0.833·17-s + 0.977i·19-s + 3.34i·20-s + 1.03i·23-s − 1.88·25-s + 0.453i·26-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=((−0.972+0.231i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s+1/2) L(s)=((−0.972+0.231i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: −0.972+0.231i-0.972 + 0.231i
Analytic conductor: 8.695708.69570
Root analytic conductor: 2.948842.94884
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1089(1088,⋅)\chi_{1089} (1088, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1089, ( :1/2), −0.972+0.231i)(2,\ 1089,\ (\ :1/2),\ -0.972 + 0.231i)

Particular Values

L(1)L(1) ≈\approx 0.29020859540.2902085954
L(12)L(\frac12) ≈\approx 0.29020859540.2902085954
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)
bad3 1 1
11 1 1
good2 1+2.43T+2T2 1 + 2.43T + 2T^{2}
5 1−3.79iT−5T2 1 - 3.79iT - 5T^{2}
7 1−0.367iT−7T2 1 - 0.367iT - 7T^{2}
13 1+0.948iT−13T2 1 + 0.948iT - 13T^{2}
17 1+3.43T+17T2 1 + 3.43T + 17T^{2}
19 1−4.26iT−19T2 1 - 4.26iT - 19T^{2}
23 1−4.96iT−23T2 1 - 4.96iT - 23T^{2}
29 1+2.48T+29T2 1 + 2.48T + 29T^{2}
31 1−3.51T+31T2 1 - 3.51T + 31T^{2}
37 1+7.18T+37T2 1 + 7.18T + 37T^{2}
41 1+2.71T+41T2 1 + 2.71T + 41T^{2}
43 1+1.88iT−43T2 1 + 1.88iT - 43T^{2}
47 1−0.0206iT−47T2 1 - 0.0206iT - 47T^{2}
53 1+5.54iT−53T2 1 + 5.54iT - 53T^{2}
59 1+6.62iT−59T2 1 + 6.62iT - 59T^{2}
61 1−9.75iT−61T2 1 - 9.75iT - 61T^{2}
67 1−4.46T+67T2 1 - 4.46T + 67T^{2}
71 1+10.3iT−71T2 1 + 10.3iT - 71T^{2}
73 1−4.39iT−73T2 1 - 4.39iT - 73T^{2}
79 1−10.9iT−79T2 1 - 10.9iT - 79T^{2}
83 1+9.18T+83T2 1 + 9.18T + 83T^{2}
89 1+3.04iT−89T2 1 + 3.04iT - 89T^{2}
97 1+15.0T+97T2 1 + 15.0T + 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

−10.19116671087564725155100742111, −9.659825728375226606288910237872, −8.660251266215841459501006329769, −7.88087598729276182783857001913, −7.12191493927450181128236273420, −6.60631919215030036856128197388, −5.63560395915246864087765718958, −3.74460925830925112486475350559, −2.71084964315007959743580427594, −1.76134495542290762331392165882, 0.23600968729111307105706260439, 1.30492980399026857215308234693, 2.41893417288374219102995254190, 4.24956708819211652331820602001, 5.10226307252524744673554936567, 6.36938163211037379726028965745, 7.20852008654909486287121125540, 8.157391774747920734857841004557, 8.816613948352231354422520921037, 9.098465762272512833095743770795

Graph of the ZZ-function along the critical line