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Properties

Label 2-546-1.1-c7-0-67
Degree 22
Conductor 546546
Sign −1-1
Analytic cond. 170.562170.562
Root an. cond. 13.059913.0599
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 27·3-s + 64·4-s − 135·5-s − 216·6-s + 343·7-s + 512·8-s + 729·9-s − 1.08e3·10-s + 4.19e3·11-s − 1.72e3·12-s + 2.19e3·13-s + 2.74e3·14-s + 3.64e3·15-s + 4.09e3·16-s − 1.27e4·17-s + 5.83e3·18-s − 2.82e4·19-s − 8.64e3·20-s − 9.26e3·21-s + 3.35e4·22-s − 5.70e4·23-s − 1.38e4·24-s − 5.99e4·25-s + 1.75e4·26-s − 1.96e4·27-s + 2.19e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.482·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.341·10-s + 0.950·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.278·15-s + 1/4·16-s − 0.628·17-s + 0.235·18-s − 0.943·19-s − 0.241·20-s − 0.218·21-s + 0.672·22-s − 0.977·23-s − 0.204·24-s − 0.766·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(546s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(546s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, 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: −1-1
Analytic conductor: 170.562170.562
Root analytic conductor: 13.059913.0599
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 546, ( :7/2), −1)(2,\ 546,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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−p3T 1 - p^{3} T
3 1+p3T 1 + p^{3} T
7 1−p3T 1 - p^{3} T
13 1−p3T 1 - p^{3} T
good5 1+27pT+p7T2 1 + 27 p T + p^{7} T^{2}
11 1−4197T+p7T2 1 - 4197 T + p^{7} T^{2}
17 1+12735T+p7T2 1 + 12735 T + p^{7} T^{2}
19 1+28213T+p7T2 1 + 28213 T + p^{7} T^{2}
23 1+57039T+p7T2 1 + 57039 T + p^{7} T^{2}
29 1+10269T+p7T2 1 + 10269 T + p^{7} T^{2}
31 1−98276T+p7T2 1 - 98276 T + p^{7} T^{2}
37 1+352033T+p7T2 1 + 352033 T + p^{7} T^{2}
41 1−473172T+p7T2 1 - 473172 T + p^{7} T^{2}
43 1−891395T+p7T2 1 - 891395 T + p^{7} T^{2}
47 1−684984T+p7T2 1 - 684984 T + p^{7} T^{2}
53 1−271002T+p7T2 1 - 271002 T + p^{7} T^{2}
59 1+954024T+p7T2 1 + 954024 T + p^{7} T^{2}
61 1−3197159T+p7T2 1 - 3197159 T + p^{7} T^{2}
67 1+2902018T+p7T2 1 + 2902018 T + p^{7} T^{2}
71 1+4599768T+p7T2 1 + 4599768 T + p^{7} T^{2}
73 1−1277111T+p7T2 1 - 1277111 T + p^{7} T^{2}
79 1+1928494T+p7T2 1 + 1928494 T + p^{7} T^{2}
83 1+355674T+p7T2 1 + 355674 T + p^{7} T^{2}
89 1+217854T+p7T2 1 + 217854 T + p^{7} T^{2}
97 1+4400758T+p7T2 1 + 4400758 T + p^{7} 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

−9.214274797297948456064618775694, −8.213663247073090905588359620909, −7.21830599901927250170162794147, −6.33534133821146902410671781088, −5.59675108818321936155218270203, −4.24319206421528847620621857468, −4.02066188337132426350134019025, −2.39825111046784900767692870102, −1.29806096478780927900362405597, 0, 1.29806096478780927900362405597, 2.39825111046784900767692870102, 4.02066188337132426350134019025, 4.24319206421528847620621857468, 5.59675108818321936155218270203, 6.33534133821146902410671781088, 7.21830599901927250170162794147, 8.213663247073090905588359620909, 9.214274797297948456064618775694

Graph of the ZZ-function along the critical line