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Properties

Label 2-891-1.1-c1-0-12
Degree 22
Conductor 891891
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.167·2-s − 1.97·4-s + 2.16·5-s + 3.80·7-s + 0.665·8-s − 0.362·10-s + 11-s − 0.167·13-s − 0.637·14-s + 3.83·16-s − 1.13·17-s − 4.63·19-s − 4.27·20-s − 0.167·22-s + 4.13·23-s − 0.302·25-s + 0.0280·26-s − 7.50·28-s + 9.97·29-s − 1.02·31-s − 1.97·32-s + 0.190·34-s + 8.24·35-s − 4.94·37-s + 0.776·38-s + 1.44·40-s − 2.97·41-s + ⋯
L(s)  = 1  − 0.118·2-s − 0.985·4-s + 0.969·5-s + 1.43·7-s + 0.235·8-s − 0.114·10-s + 0.301·11-s − 0.0464·13-s − 0.170·14-s + 0.958·16-s − 0.276·17-s − 1.06·19-s − 0.955·20-s − 0.0357·22-s + 0.863·23-s − 0.0604·25-s + 0.00549·26-s − 1.41·28-s + 1.85·29-s − 0.184·31-s − 0.348·32-s + 0.0327·34-s + 1.39·35-s − 0.812·37-s + 0.125·38-s + 0.227·40-s − 0.464·41-s + ⋯

Functional equation

Λ(s)=(891s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(891s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\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    =    34⋅113^{4} \cdot 11
Sign: 11
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: 00
Selberg data: (2, 891, ( :1/2), 1)(2,\ 891,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6709879051.670987905
L(12)L(\frac12) ≈\approx 1.6709879051.670987905
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−T 1 - T
good2 1+0.167T+2T2 1 + 0.167T + 2T^{2}
5 1−2.16T+5T2 1 - 2.16T + 5T^{2}
7 1−3.80T+7T2 1 - 3.80T + 7T^{2}
13 1+0.167T+13T2 1 + 0.167T + 13T^{2}
17 1+1.13T+17T2 1 + 1.13T + 17T^{2}
19 1+4.63T+19T2 1 + 4.63T + 19T^{2}
23 1−4.13T+23T2 1 - 4.13T + 23T^{2}
29 1−9.97T+29T2 1 - 9.97T + 29T^{2}
31 1+1.02T+31T2 1 + 1.02T + 31T^{2}
37 1+4.94T+37T2 1 + 4.94T + 37T^{2}
41 1+2.97T+41T2 1 + 2.97T + 41T^{2}
43 1−5.80T+43T2 1 - 5.80T + 43T^{2}
47 1−6.63T+47T2 1 - 6.63T + 47T^{2}
53 1−11.2T+53T2 1 - 11.2T + 53T^{2}
59 1−14.4T+59T2 1 - 14.4T + 59T^{2}
61 1+2.80T+61T2 1 + 2.80T + 61T^{2}
67 1+14.4T+67T2 1 + 14.4T + 67T^{2}
71 1+12.5T+71T2 1 + 12.5T + 71T^{2}
73 1−14.5T+73T2 1 - 14.5T + 73T^{2}
79 1−7.44T+79T2 1 - 7.44T + 79T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1−8.02T+89T2 1 - 8.02T + 89T^{2}
97 1−6.27T+97T2 1 - 6.27T + 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

−10.23551248554371332264292310984, −9.034659748140168951726846094451, −8.697287105503682778645045988066, −7.80013618321647868680495718775, −6.64549732038620289286041608268, −5.54779331247304747450148310924, −4.85238249206593321434519936238, −4.04088199767727839339744429041, −2.35353106777765548274119533919, −1.18072091599964512308474894277, 1.18072091599964512308474894277, 2.35353106777765548274119533919, 4.04088199767727839339744429041, 4.85238249206593321434519936238, 5.54779331247304747450148310924, 6.64549732038620289286041608268, 7.80013618321647868680495718775, 8.697287105503682778645045988066, 9.034659748140168951726846094451, 10.23551248554371332264292310984

Graph of the ZZ-function along the critical line