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Properties

Label 2-33e2-1.1-c1-0-25
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  + 2.64·2-s + 5.00·4-s − 2.64·5-s + 2·7-s + 7.93·8-s − 7.00·10-s + 5·13-s + 5.29·14-s + 11.0·16-s + 2.64·17-s − 13.2·20-s − 5.29·23-s + 2.00·25-s + 13.2·26-s + 10.0·28-s − 7.93·29-s + 4·31-s + 13.2·32-s + 7.00·34-s − 5.29·35-s − 3·37-s − 21.0·40-s + 2.64·41-s + 6·43-s − 14.0·46-s + 10.5·47-s − 3·49-s + ⋯
L(s)  = 1  + 1.87·2-s + 2.50·4-s − 1.18·5-s + 0.755·7-s + 2.80·8-s − 2.21·10-s + 1.38·13-s + 1.41·14-s + 2.75·16-s + 0.641·17-s − 2.95·20-s − 1.10·23-s + 0.400·25-s + 2.59·26-s + 1.88·28-s − 1.47·29-s + 0.718·31-s + 2.33·32-s + 1.20·34-s − 0.894·35-s − 0.493·37-s − 3.32·40-s + 0.413·41-s + 0.914·43-s − 2.06·46-s + 1.54·47-s − 0.428·49-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 4.7666517124.766651712
L(12)L(\frac12) ≈\approx 4.7666517124.766651712
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 1−2.64T+2T2 1 - 2.64T + 2T^{2}
5 1+2.64T+5T2 1 + 2.64T + 5T^{2}
7 1−2T+7T2 1 - 2T + 7T^{2}
13 1−5T+13T2 1 - 5T + 13T^{2}
17 1−2.64T+17T2 1 - 2.64T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+5.29T+23T2 1 + 5.29T + 23T^{2}
29 1+7.93T+29T2 1 + 7.93T + 29T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+3T+37T2 1 + 3T + 37T^{2}
41 1−2.64T+41T2 1 - 2.64T + 41T^{2}
43 1−6T+43T2 1 - 6T + 43T^{2}
47 1−10.5T+47T2 1 - 10.5T + 47T^{2}
53 1−7.93T+53T2 1 - 7.93T + 53T^{2}
59 1+5.29T+59T2 1 + 5.29T + 59T^{2}
61 1+6T+61T2 1 + 6T + 61T^{2}
67 1+10T+67T2 1 + 10T + 67T^{2}
71 1+15.8T+71T2 1 + 15.8T + 71T^{2}
73 1+10T+73T2 1 + 10T + 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+7.93T+89T2 1 + 7.93T + 89T^{2}
97 1−5T+97T2 1 - 5T + 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

−10.42050543841446949014843697780, −8.779446533560007743599347167638, −7.77594574784300739537262892995, −7.35922998547033483473724572447, −6.08939320500831878326768482673, −5.56139704051060845600743152726, −4.29707997991328737318830135800, −3.99480843407406712831031412201, −3.01381741469607397174690710148, −1.59089441543721140954265628235, 1.59089441543721140954265628235, 3.01381741469607397174690710148, 3.99480843407406712831031412201, 4.29707997991328737318830135800, 5.56139704051060845600743152726, 6.08939320500831878326768482673, 7.35922998547033483473724572447, 7.77594574784300739537262892995, 8.779446533560007743599347167638, 10.42050543841446949014843697780

Graph of the ZZ-function along the critical line