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Properties

Label 2-33e2-1.1-c1-0-11
Degree 22
Conductor 10891089
Sign 11
Analytic cond. 8.695708.69570
Root an. cond. 2.948842.94884
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 0.618·4-s + 0.381·5-s + 3·7-s + 2.23·8-s − 0.618·10-s + 6.23·13-s − 4.85·14-s − 4.85·16-s + 0.618·17-s − 0.854·19-s + 0.236·20-s + 5.47·23-s − 4.85·25-s − 10.0·26-s + 1.85·28-s + 4.47·29-s − 3.85·31-s + 3.38·32-s − 1.00·34-s + 1.14·35-s − 4.23·37-s + 1.38·38-s + 0.854·40-s − 5.94·41-s + 1.76·43-s − 8.85·46-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.309·4-s + 0.170·5-s + 1.13·7-s + 0.790·8-s − 0.195·10-s + 1.72·13-s − 1.29·14-s − 1.21·16-s + 0.149·17-s − 0.195·19-s + 0.0527·20-s + 1.14·23-s − 0.970·25-s − 1.97·26-s + 0.350·28-s + 0.830·29-s − 0.692·31-s + 0.597·32-s − 0.171·34-s + 0.193·35-s − 0.696·37-s + 0.224·38-s + 0.135·40-s − 0.928·41-s + 0.268·43-s − 1.30·46-s + ⋯

Functional equation

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

Invariants

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

Particular Values

L(1)L(1) ≈\approx 1.0836172641.083617264
L(12)L(\frac12) ≈\approx 1.0836172641.083617264
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+1.61T+2T2 1 + 1.61T + 2T^{2}
5 1−0.381T+5T2 1 - 0.381T + 5T^{2}
7 1−3T+7T2 1 - 3T + 7T^{2}
13 1−6.23T+13T2 1 - 6.23T + 13T^{2}
17 1−0.618T+17T2 1 - 0.618T + 17T^{2}
19 1+0.854T+19T2 1 + 0.854T + 19T^{2}
23 1−5.47T+23T2 1 - 5.47T + 23T^{2}
29 1−4.47T+29T2 1 - 4.47T + 29T^{2}
31 1+3.85T+31T2 1 + 3.85T + 31T^{2}
37 1+4.23T+37T2 1 + 4.23T + 37T^{2}
41 1+5.94T+41T2 1 + 5.94T + 41T^{2}
43 1−1.76T+43T2 1 - 1.76T + 43T^{2}
47 1−0.618T+47T2 1 - 0.618T + 47T^{2}
53 1−7.38T+53T2 1 - 7.38T + 53T^{2}
59 1−5.32T+59T2 1 - 5.32T + 59T^{2}
61 1−1.14T+61T2 1 - 1.14T + 61T^{2}
67 1−10.5T+67T2 1 - 10.5T + 67T^{2}
71 1+14.5T+71T2 1 + 14.5T + 71T^{2}
73 1−1.23T+73T2 1 - 1.23T + 73T^{2}
79 1−0.527T+79T2 1 - 0.527T + 79T^{2}
83 1−12.7T+83T2 1 - 12.7T + 83T^{2}
89 1+9.47T+89T2 1 + 9.47T + 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

−9.810594438027469134619056506416, −8.726879803860141242320017408954, −8.549773179661251261034003107670, −7.66069687244334961841222307311, −6.77549609985789858096673811816, −5.63447000050821234618991322262, −4.68821999317995111248257246208, −3.63330394351187429002130763804, −1.94040847984149739027423933471, −1.03899719477631895175166871998, 1.03899719477631895175166871998, 1.94040847984149739027423933471, 3.63330394351187429002130763804, 4.68821999317995111248257246208, 5.63447000050821234618991322262, 6.77549609985789858096673811816, 7.66069687244334961841222307311, 8.549773179661251261034003107670, 8.726879803860141242320017408954, 9.810594438027469134619056506416

Graph of the ZZ-function along the critical line