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Properties

Label 2-459-1.1-c1-0-15
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  − 0.618·2-s − 1.61·4-s − 0.381·5-s + 1.85·7-s + 2.23·8-s + 0.236·10-s − 6.23·11-s − 1.23·13-s − 1.14·14-s + 1.85·16-s − 17-s + 3·19-s + 0.618·20-s + 3.85·22-s − 7.09·23-s − 4.85·25-s + 0.763·26-s − 3·28-s − 3.76·29-s − 0.236·31-s − 5.61·32-s + 0.618·34-s − 0.708·35-s − 8.47·37-s − 1.85·38-s − 0.854·40-s − 8.61·41-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.809·4-s − 0.170·5-s + 0.700·7-s + 0.790·8-s + 0.0746·10-s − 1.88·11-s − 0.342·13-s − 0.306·14-s + 0.463·16-s − 0.242·17-s + 0.688·19-s + 0.138·20-s + 0.821·22-s − 1.47·23-s − 0.970·25-s + 0.149·26-s − 0.566·28-s − 0.698·29-s − 0.0423·31-s − 0.993·32-s + 0.105·34-s − 0.119·35-s − 1.39·37-s − 0.300·38-s − 0.135·40-s − 1.34·41-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+0.618T+2T2 1 + 0.618T + 2T^{2}
5 1+0.381T+5T2 1 + 0.381T + 5T^{2}
7 1−1.85T+7T2 1 - 1.85T + 7T^{2}
11 1+6.23T+11T2 1 + 6.23T + 11T^{2}
13 1+1.23T+13T2 1 + 1.23T + 13T^{2}
19 1−3T+19T2 1 - 3T + 19T^{2}
23 1+7.09T+23T2 1 + 7.09T + 23T^{2}
29 1+3.76T+29T2 1 + 3.76T + 29T^{2}
31 1+0.236T+31T2 1 + 0.236T + 31T^{2}
37 1+8.47T+37T2 1 + 8.47T + 37T^{2}
41 1+8.61T+41T2 1 + 8.61T + 41T^{2}
43 1+1.85T+43T2 1 + 1.85T + 43T^{2}
47 1−4.32T+47T2 1 - 4.32T + 47T^{2}
53 1−5.32T+53T2 1 - 5.32T + 53T^{2}
59 1−7.09T+59T2 1 - 7.09T + 59T^{2}
61 1−15.0T+61T2 1 - 15.0T + 61T^{2}
67 1+9.85T+67T2 1 + 9.85T + 67T^{2}
71 1−6.23T+71T2 1 - 6.23T + 71T^{2}
73 1+2.52T+73T2 1 + 2.52T + 73T^{2}
79 1+4.85T+79T2 1 + 4.85T + 79T^{2}
83 1−6.52T+83T2 1 - 6.52T + 83T^{2}
89 1−9.32T+89T2 1 - 9.32T + 89T^{2}
97 1−0.909T+97T2 1 - 0.909T + 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

−10.30861467391288214749127082394, −9.916728527331014851382104552332, −8.630960899456274033118945197119, −7.988638855078182716717421417526, −7.34687233561946531043195396841, −5.54846567537103418410337763388, −4.95684774125784851642822673312, −3.73077871325009449494545283471, −2.05437589228643690193884427190, 0, 2.05437589228643690193884427190, 3.73077871325009449494545283471, 4.95684774125784851642822673312, 5.54846567537103418410337763388, 7.34687233561946531043195396841, 7.988638855078182716717421417526, 8.630960899456274033118945197119, 9.916728527331014851382104552332, 10.30861467391288214749127082394

Graph of the ZZ-function along the critical line