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Properties

Label 2-459-1.1-c1-0-21
Degree 22
Conductor 459459
Sign −1-1
Analytic cond. 3.665133.66513
Root an. cond. 1.914451.91445
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  + 1.61·2-s + 0.618·4-s − 2.61·5-s − 4.85·7-s − 2.23·8-s − 4.23·10-s − 1.76·11-s + 3.23·13-s − 7.85·14-s − 4.85·16-s − 17-s + 3·19-s − 1.61·20-s − 2.85·22-s + 4.09·23-s + 1.85·25-s + 5.23·26-s − 3.00·28-s − 8.23·29-s + 4.23·31-s − 3.38·32-s − 1.61·34-s + 12.7·35-s + 0.472·37-s + 4.85·38-s + 5.85·40-s − 6.38·41-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.309·4-s − 1.17·5-s − 1.83·7-s − 0.790·8-s − 1.33·10-s − 0.531·11-s + 0.897·13-s − 2.09·14-s − 1.21·16-s − 0.242·17-s + 0.688·19-s − 0.361·20-s − 0.608·22-s + 0.852·23-s + 0.370·25-s + 1.02·26-s − 0.566·28-s − 1.52·29-s + 0.760·31-s − 0.597·32-s − 0.277·34-s + 2.14·35-s + 0.0776·37-s + 0.787·38-s + 0.925·40-s − 0.996·41-s + ⋯

Functional equation

Λ(s)=(459s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(459s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 459459    =    33⋅173^{3} \cdot 17
Sign: −1-1
Analytic conductor: 3.665133.66513
Root analytic conductor: 1.914451.91445
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 459, ( :1/2), −1)(2,\ 459,\ (\ :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(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
17 1+T 1 + T
good2 1−1.61T+2T2 1 - 1.61T + 2T^{2}
5 1+2.61T+5T2 1 + 2.61T + 5T^{2}
7 1+4.85T+7T2 1 + 4.85T + 7T^{2}
11 1+1.76T+11T2 1 + 1.76T + 11T^{2}
13 1−3.23T+13T2 1 - 3.23T + 13T^{2}
19 1−3T+19T2 1 - 3T + 19T^{2}
23 1−4.09T+23T2 1 - 4.09T + 23T^{2}
29 1+8.23T+29T2 1 + 8.23T + 29T^{2}
31 1−4.23T+31T2 1 - 4.23T + 31T^{2}
37 1−0.472T+37T2 1 - 0.472T + 37T^{2}
41 1+6.38T+41T2 1 + 6.38T + 41T^{2}
43 1−4.85T+43T2 1 - 4.85T + 43T^{2}
47 1+11.3T+47T2 1 + 11.3T + 47T^{2}
53 1+10.3T+53T2 1 + 10.3T + 53T^{2}
59 1+4.09T+59T2 1 + 4.09T + 59T^{2}
61 1+14.0T+61T2 1 + 14.0T + 61T^{2}
67 1+3.14T+67T2 1 + 3.14T + 67T^{2}
71 1−1.76T+71T2 1 - 1.76T + 71T^{2}
73 1+11.4T+73T2 1 + 11.4T + 73T^{2}
79 1−1.85T+79T2 1 - 1.85T + 79T^{2}
83 1−15.4T+83T2 1 - 15.4T + 83T^{2}
89 1+6.32T+89T2 1 + 6.32T + 89T^{2}
97 1−12.0T+97T2 1 - 12.0T + 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

−10.90348513533437078225182270530, −9.656793169128973856572005684850, −8.894011005100265104473870482758, −7.66985811191610761036146548009, −6.62670946638433761725637379426, −5.84613034863095469794746401520, −4.62583440749178726044275294502, −3.49921530746824352302871150473, −3.13945368839014386283856455216, 0, 3.13945368839014386283856455216, 3.49921530746824352302871150473, 4.62583440749178726044275294502, 5.84613034863095469794746401520, 6.62670946638433761725637379426, 7.66985811191610761036146548009, 8.894011005100265104473870482758, 9.656793169128973856572005684850, 10.90348513533437078225182270530

Graph of the ZZ-function along the critical line