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Properties

Label 2-546-1.1-c5-0-26
Degree 22
Conductor 546546
Sign 11
Analytic cond. 87.569587.5695
Root an. cond. 9.357869.35786
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 9·3-s + 16·4-s + 93·5-s − 36·6-s − 49·7-s + 64·8-s + 81·9-s + 372·10-s − 302·11-s − 144·12-s − 169·13-s − 196·14-s − 837·15-s + 256·16-s − 488·17-s + 324·18-s + 2.05e3·19-s + 1.48e3·20-s + 441·21-s − 1.20e3·22-s + 59·23-s − 576·24-s + 5.52e3·25-s − 676·26-s − 729·27-s − 784·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.66·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.17·10-s − 0.752·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.960·15-s + 1/4·16-s − 0.409·17-s + 0.235·18-s + 1.30·19-s + 0.831·20-s + 0.218·21-s − 0.532·22-s + 0.0232·23-s − 0.204·24-s + 1.76·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(546s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(546s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 546546    =    2⋅3⋅7⋅132 \cdot 3 \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 87.569587.5695
Root analytic conductor: 9.357869.35786
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 546, ( :5/2), 1)(2,\ 546,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.9153606443.915360644
L(12)L(\frac12) ≈\approx 3.9153606443.915360644
L(72)L(\frac{7}{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)
bad2 1−p2T 1 - p^{2} T
3 1+p2T 1 + p^{2} T
7 1+p2T 1 + p^{2} T
13 1+p2T 1 + p^{2} T
good5 1−93T+p5T2 1 - 93 T + p^{5} T^{2}
11 1+302T+p5T2 1 + 302 T + p^{5} T^{2}
17 1+488T+p5T2 1 + 488 T + p^{5} T^{2}
19 1−2053T+p5T2 1 - 2053 T + p^{5} T^{2}
23 1−59T+p5T2 1 - 59 T + p^{5} T^{2}
29 1−5871T+p5T2 1 - 5871 T + p^{5} T^{2}
31 1−3861T+p5T2 1 - 3861 T + p^{5} T^{2}
37 1−12388T+p5T2 1 - 12388 T + p^{5} T^{2}
41 1−2602T+p5T2 1 - 2602 T + p^{5} T^{2}
43 1+14221T+p5T2 1 + 14221 T + p^{5} T^{2}
47 1+21645T+p5T2 1 + 21645 T + p^{5} T^{2}
53 1+7781T+p5T2 1 + 7781 T + p^{5} T^{2}
59 1−19072T+p5T2 1 - 19072 T + p^{5} T^{2}
61 1−13954T+p5T2 1 - 13954 T + p^{5} T^{2}
67 1−2694T+p5T2 1 - 2694 T + p^{5} T^{2}
71 1−82032T+p5T2 1 - 82032 T + p^{5} T^{2}
73 1−6503T+p5T2 1 - 6503 T + p^{5} T^{2}
79 1−28535T+p5T2 1 - 28535 T + p^{5} T^{2}
83 1−15019T+p5T2 1 - 15019 T + p^{5} T^{2}
89 1−41979T+p5T2 1 - 41979 T + p^{5} T^{2}
97 1+57405T+p5T2 1 + 57405 T + p^{5} T^{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

−9.944785592925614786115180717441, −9.612688043862781438649632375354, −8.134987767750928694307930019746, −6.84334186655392320241116812926, −6.22550362869951504678123493488, −5.36230832545086776772802267502, −4.74977725312544173653801633961, −3.08019766612721338552325304671, −2.18849388548179538541415294718, −0.927872047644358692894540287180, 0.927872047644358692894540287180, 2.18849388548179538541415294718, 3.08019766612721338552325304671, 4.74977725312544173653801633961, 5.36230832545086776772802267502, 6.22550362869951504678123493488, 6.84334186655392320241116812926, 8.134987767750928694307930019746, 9.612688043862781438649632375354, 9.944785592925614786115180717441

Graph of the ZZ-function along the critical line