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Properties

Label 2-33e2-33.32-c1-0-6
Degree 22
Conductor 10891089
Sign 0.384−0.923i0.384 - 0.923i
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  − 0.688·2-s − 1.52·4-s + 0.0401i·5-s − 0.246i·7-s + 2.42·8-s − 0.0276i·10-s + 2.30i·13-s + 0.169i·14-s + 1.38·16-s − 4.27·17-s − 6.20i·19-s − 0.0612i·20-s + 6.79i·23-s + 4.99·25-s − 1.58i·26-s + ⋯
L(s)  = 1  − 0.486·2-s − 0.763·4-s + 0.0179i·5-s − 0.0932i·7-s + 0.858·8-s − 0.00872i·10-s + 0.638i·13-s + 0.0454i·14-s + 0.345·16-s − 1.03·17-s − 1.42i·19-s − 0.0136i·20-s + 1.41i·23-s + 0.999·25-s − 0.310i·26-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=((0.384−0.923i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s+1/2) L(s)=((0.384−0.923i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: 0.384−0.923i0.384 - 0.923i
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), 0.384−0.923i)(2,\ 1089,\ (\ :1/2),\ 0.384 - 0.923i)

Particular Values

L(1)L(1) ≈\approx 0.79418044660.7941804466
L(12)L(\frac12) ≈\approx 0.79418044660.7941804466
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+0.688T+2T2 1 + 0.688T + 2T^{2}
5 1−0.0401iT−5T2 1 - 0.0401iT - 5T^{2}
7 1+0.246iT−7T2 1 + 0.246iT - 7T^{2}
13 1−2.30iT−13T2 1 - 2.30iT - 13T^{2}
17 1+4.27T+17T2 1 + 4.27T + 17T^{2}
19 1+6.20iT−19T2 1 + 6.20iT - 19T^{2}
23 1−6.79iT−23T2 1 - 6.79iT - 23T^{2}
29 1+5.59T+29T2 1 + 5.59T + 29T^{2}
31 1−4.79T+31T2 1 - 4.79T + 31T^{2}
37 1+4.03T+37T2 1 + 4.03T + 37T^{2}
41 1−9.60T+41T2 1 - 9.60T + 41T^{2}
43 1−1.03iT−43T2 1 - 1.03iT - 43T^{2}
47 1−11.1iT−47T2 1 - 11.1iT - 47T^{2}
53 1−8.96iT−53T2 1 - 8.96iT - 53T^{2}
59 1−2.78iT−59T2 1 - 2.78iT - 59T^{2}
61 1−8.48iT−61T2 1 - 8.48iT - 61T^{2}
67 1−7.94T+67T2 1 - 7.94T + 67T^{2}
71 1−3.32iT−71T2 1 - 3.32iT - 71T^{2}
73 1−11.8iT−73T2 1 - 11.8iT - 73T^{2}
79 1−3.01iT−79T2 1 - 3.01iT - 79T^{2}
83 1+5.29T+83T2 1 + 5.29T + 83T^{2}
89 1−8.54iT−89T2 1 - 8.54iT - 89T^{2}
97 1+3.02T+97T2 1 + 3.02T + 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

−9.749590656697379537345782249528, −9.151847031324253111812193096987, −8.661944624461309309707127280918, −7.53239586995472275178768725296, −6.93045106611899315665111024350, −5.71183822392350844525876943205, −4.69493841996800004765734577326, −4.03858853264800273785022775858, −2.61713481083898514510419594548, −1.11819915132474521782161659814, 0.51408885130907543948359990438, 2.07359307152150981892691724188, 3.53231076987696099132894774539, 4.48285572926991813837212151585, 5.35012247395157256996385435979, 6.37408112398852213898326472885, 7.40735584071940610329828522591, 8.341813322465757819307096544023, 8.740319946165314148553589741992, 9.709259270341238189946421873510

Graph of the ZZ-function along the critical line