Properties

Label 2-891-1.1-c1-0-34
Degree 22
Conductor 891891
Sign 1-1
Analytic cond. 7.114677.11467
Root an. cond. 2.667332.66733
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.347·2-s − 1.87·4-s + 0.652·5-s + 0.532·7-s − 1.34·8-s + 0.226·10-s − 11-s − 2.30·13-s + 0.184·14-s + 3.29·16-s − 4.41·17-s − 4.18·19-s − 1.22·20-s − 0.347·22-s + 1.41·23-s − 4.57·25-s − 0.800·26-s − 28-s + 6.35·29-s − 2.16·31-s + 3.83·32-s − 1.53·34-s + 0.347·35-s − 3.16·37-s − 1.45·38-s − 0.879·40-s − 9.31·41-s + ⋯
L(s)  = 1  + 0.245·2-s − 0.939·4-s + 0.291·5-s + 0.201·7-s − 0.476·8-s + 0.0716·10-s − 0.301·11-s − 0.639·13-s + 0.0493·14-s + 0.822·16-s − 1.06·17-s − 0.960·19-s − 0.274·20-s − 0.0740·22-s + 0.294·23-s − 0.914·25-s − 0.157·26-s − 0.188·28-s + 1.18·29-s − 0.388·31-s + 0.678·32-s − 0.262·34-s + 0.0587·35-s − 0.519·37-s − 0.235·38-s − 0.139·40-s − 1.45·41-s + ⋯

Functional equation

Λ(s)=(891s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(891s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 891891    =    34113^{4} \cdot 11
Sign: 1-1
Analytic conductor: 7.114677.11467
Root analytic conductor: 2.667332.66733
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 891, ( :1/2), 1)(2,\ 891,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
11 1+T 1 + T
good2 10.347T+2T2 1 - 0.347T + 2T^{2}
5 10.652T+5T2 1 - 0.652T + 5T^{2}
7 10.532T+7T2 1 - 0.532T + 7T^{2}
13 1+2.30T+13T2 1 + 2.30T + 13T^{2}
17 1+4.41T+17T2 1 + 4.41T + 17T^{2}
19 1+4.18T+19T2 1 + 4.18T + 19T^{2}
23 11.41T+23T2 1 - 1.41T + 23T^{2}
29 16.35T+29T2 1 - 6.35T + 29T^{2}
31 1+2.16T+31T2 1 + 2.16T + 31T^{2}
37 1+3.16T+37T2 1 + 3.16T + 37T^{2}
41 1+9.31T+41T2 1 + 9.31T + 41T^{2}
43 1+12.2T+43T2 1 + 12.2T + 43T^{2}
47 1+3.86T+47T2 1 + 3.86T + 47T^{2}
53 112.6T+53T2 1 - 12.6T + 53T^{2}
59 1+3.23T+59T2 1 + 3.23T + 59T^{2}
61 1+8.53T+61T2 1 + 8.53T + 61T^{2}
67 14.96T+67T2 1 - 4.96T + 67T^{2}
71 19.98T+71T2 1 - 9.98T + 71T^{2}
73 1+7.49T+73T2 1 + 7.49T + 73T^{2}
79 1+4.87T+79T2 1 + 4.87T + 79T^{2}
83 1+4.59T+83T2 1 + 4.59T + 83T^{2}
89 1+16.9T+89T2 1 + 16.9T + 89T^{2}
97 112.6T+97T2 1 - 12.6T + 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.813528084012545849001413704947, −8.715451400683312969934453774565, −8.322185163147674644761020573079, −7.06579729254783944256956272214, −6.14106593976144323564404634477, −5.05887063621192910141549236423, −4.50617322891500742157460201944, −3.31908492930294815809901345461, −1.97679894025303810451404739737, 0, 1.97679894025303810451404739737, 3.31908492930294815809901345461, 4.50617322891500742157460201944, 5.05887063621192910141549236423, 6.14106593976144323564404634477, 7.06579729254783944256956272214, 8.322185163147674644761020573079, 8.715451400683312969934453774565, 9.813528084012545849001413704947

Graph of the ZZ-function along the critical line