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Properties

Label 2-28-1.1-c5-0-0
Degree 22
Conductor 2828
Sign 11
Analytic cond. 4.490744.49074
Root an. cond. 2.119132.11913
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  + 26·3-s + 16·5-s − 49·7-s + 433·9-s + 8·11-s + 684·13-s + 416·15-s − 2.21e3·17-s − 2.69e3·19-s − 1.27e3·21-s + 3.34e3·23-s − 2.86e3·25-s + 4.94e3·27-s − 3.25e3·29-s + 4.78e3·31-s + 208·33-s − 784·35-s − 1.14e4·37-s + 1.77e4·39-s + 1.33e4·41-s − 928·43-s + 6.92e3·45-s + 1.21e3·47-s + 2.40e3·49-s − 5.76e4·51-s + 1.31e4·53-s + 128·55-s + ⋯
L(s)  = 1  + 1.66·3-s + 0.286·5-s − 0.377·7-s + 1.78·9-s + 0.0199·11-s + 1.12·13-s + 0.477·15-s − 1.86·17-s − 1.71·19-s − 0.630·21-s + 1.31·23-s − 0.918·25-s + 1.30·27-s − 0.718·29-s + 0.894·31-s + 0.0332·33-s − 0.108·35-s − 1.37·37-s + 1.87·39-s + 1.24·41-s − 0.0765·43-s + 0.510·45-s + 0.0800·47-s + 1/7·49-s − 3.10·51-s + 0.641·53-s + 0.00570·55-s + ⋯

Functional equation

Λ(s)=(28s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(28s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2828    =    22⋅72^{2} \cdot 7
Sign: 11
Analytic conductor: 4.490744.49074
Root analytic conductor: 2.119132.11913
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 28, ( :5/2), 1)(2,\ 28,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.3758745872.375874587
L(12)L(\frac12) ≈\approx 2.3758745872.375874587
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 1
7 1+p2T 1 + p^{2} T
good3 1−26T+p5T2 1 - 26 T + p^{5} T^{2}
5 1−16T+p5T2 1 - 16 T + p^{5} T^{2}
11 1−8T+p5T2 1 - 8 T + p^{5} T^{2}
13 1−684T+p5T2 1 - 684 T + p^{5} T^{2}
17 1+2218T+p5T2 1 + 2218 T + p^{5} T^{2}
19 1+142pT+p5T2 1 + 142 p T + p^{5} T^{2}
23 1−3344T+p5T2 1 - 3344 T + p^{5} T^{2}
29 1+3254T+p5T2 1 + 3254 T + p^{5} T^{2}
31 1−4788T+p5T2 1 - 4788 T + p^{5} T^{2}
37 1+310pT+p5T2 1 + 310 p T + p^{5} T^{2}
41 1−13350T+p5T2 1 - 13350 T + p^{5} T^{2}
43 1+928T+p5T2 1 + 928 T + p^{5} T^{2}
47 1−1212T+p5T2 1 - 1212 T + p^{5} T^{2}
53 1−13110T+p5T2 1 - 13110 T + p^{5} T^{2}
59 1−34702T+p5T2 1 - 34702 T + p^{5} T^{2}
61 1+1032T+p5T2 1 + 1032 T + p^{5} T^{2}
67 1−10108T+p5T2 1 - 10108 T + p^{5} T^{2}
71 1−62720T+p5T2 1 - 62720 T + p^{5} T^{2}
73 1+18926T+p5T2 1 + 18926 T + p^{5} T^{2}
79 1−11400T+p5T2 1 - 11400 T + p^{5} T^{2}
83 1−88958T+p5T2 1 - 88958 T + p^{5} T^{2}
89 1−19722T+p5T2 1 - 19722 T + p^{5} T^{2}
97 1−17062T+p5T2 1 - 17062 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

−15.75091300029627238404180851564, −14.96456783650792418454325624854, −13.59293478606349901112780335961, −13.04267118646930709050283206856, −10.79950993983158190076935110077, −9.206021531720503906539246669874, −8.439833948633120338922246953551, −6.67692715171034320407584043685, −3.95791997940940384003663737110, −2.24537971834612380657569247966, 2.24537971834612380657569247966, 3.95791997940940384003663737110, 6.67692715171034320407584043685, 8.439833948633120338922246953551, 9.206021531720503906539246669874, 10.79950993983158190076935110077, 13.04267118646930709050283206856, 13.59293478606349901112780335961, 14.96456783650792418454325624854, 15.75091300029627238404180851564

Graph of the ZZ-function along the critical line