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Properties

Label 2-33e2-33.32-c1-0-15
Degree 22
Conductor 10891089
Sign 0.742+0.670i0.742 + 0.670i
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  − 2.35·2-s + 3.52·4-s − 2.24i·5-s + 4.05i·7-s − 3.58·8-s + 5.28i·10-s − 4.65i·13-s − 9.52i·14-s + 1.38·16-s + 0.0762·17-s + 2.39i·19-s − 7.92i·20-s − 3.22i·23-s − 0.0541·25-s + 10.9i·26-s + ⋯
L(s)  = 1  − 1.66·2-s + 1.76·4-s − 1.00i·5-s + 1.53i·7-s − 1.26·8-s + 1.67i·10-s − 1.29i·13-s − 2.54i·14-s + 0.345·16-s + 0.0185·17-s + 0.549i·19-s − 1.77i·20-s − 0.672i·23-s − 0.0108·25-s + 2.14i·26-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=((0.742+0.670i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s+1/2) L(s)=((0.742+0.670i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: 0.742+0.670i0.742 + 0.670i
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), 0.742+0.670i)(2,\ 1089,\ (\ :1/2),\ 0.742 + 0.670i)

Particular Values

L(1)L(1) ≈\approx 0.65491003740.6549100374
L(12)L(\frac12) ≈\approx 0.65491003740.6549100374
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+2.35T+2T2 1 + 2.35T + 2T^{2}
5 1+2.24iT−5T2 1 + 2.24iT - 5T^{2}
7 1−4.05iT−7T2 1 - 4.05iT - 7T^{2}
13 1+4.65iT−13T2 1 + 4.65iT - 13T^{2}
17 1−0.0762T+17T2 1 - 0.0762T + 17T^{2}
19 1−2.39iT−19T2 1 - 2.39iT - 19T^{2}
23 1+3.22iT−23T2 1 + 3.22iT - 23T^{2}
29 1−1.83T+29T2 1 - 1.83T + 29T^{2}
31 1−1.67T+31T2 1 - 1.67T + 31T^{2}
37 1−7.26T+37T2 1 - 7.26T + 37T^{2}
41 1+8.44T+41T2 1 + 8.44T + 41T^{2}
43 1+4.28iT−43T2 1 + 4.28iT - 43T^{2}
47 1−6.21iT−47T2 1 - 6.21iT - 47T^{2}
53 1−1.22iT−53T2 1 - 1.22iT - 53T^{2}
59 1+0.580iT−59T2 1 + 0.580iT - 59T^{2}
61 1+3.78iT−61T2 1 + 3.78iT - 61T^{2}
67 1−12.9T+67T2 1 - 12.9T + 67T^{2}
71 1+1.12iT−71T2 1 + 1.12iT - 71T^{2}
73 1+13.3iT−73T2 1 + 13.3iT - 73T^{2}
79 1+0.659iT−79T2 1 + 0.659iT - 79T^{2}
83 1−10.2T+83T2 1 - 10.2T + 83T^{2}
89 1−6.58iT−89T2 1 - 6.58iT - 89T^{2}
97 1−16.4T+97T2 1 - 16.4T + 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

−9.552349670080952358068660478156, −8.888273043708382110795207780242, −8.296623360199941562867205296402, −7.85977757765401231228274141180, −6.52505248910018707448622996699, −5.67387098294094112822577897240, −4.79316296155433700274429002888, −2.98896708997841357777642368798, −1.95022436284174280585114856610, −0.66252041845139694383057129592, 0.968933882180737361786210759237, 2.22910118221352193450796195025, 3.51391495750073009017519728822, 4.62018570165626904119412241485, 6.42955551275307902723920047863, 6.93317418990346560846888670190, 7.44116879643350110344043658004, 8.322867299329349013755625697181, 9.280705063108255934688041643865, 10.01479153894407357939933273644

Graph of the ZZ-function along the critical line