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Properties

Label 2-28-1.1-c5-0-1
Degree 22
Conductor 2828
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 96·5-s + 49·7-s − 239·9-s − 720·11-s + 572·13-s + 192·15-s + 1.25e3·17-s − 94·19-s − 98·21-s + 96·23-s + 6.09e3·25-s + 964·27-s − 4.37e3·29-s − 6.24e3·31-s + 1.44e3·33-s − 4.70e3·35-s − 1.07e4·37-s − 1.14e3·39-s + 1.20e4·41-s − 9.16e3·43-s + 2.29e4·45-s − 2.58e4·47-s + 2.40e3·49-s − 2.50e3·51-s + 1.01e3·53-s + 6.91e4·55-s + ⋯
L(s)  = 1  − 0.128·3-s − 1.71·5-s + 0.377·7-s − 0.983·9-s − 1.79·11-s + 0.938·13-s + 0.220·15-s + 1.05·17-s − 0.0597·19-s − 0.0484·21-s + 0.0378·23-s + 1.94·25-s + 0.254·27-s − 0.965·29-s − 1.16·31-s + 0.230·33-s − 0.649·35-s − 1.29·37-s − 0.120·39-s + 1.11·41-s − 0.755·43-s + 1.68·45-s − 1.70·47-s + 1/7·49-s − 0.135·51-s + 0.0495·53-s + 3.08·55-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: −1-1
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: 11
Selberg data: (2, 28, ( :5/2), −1)(2,\ 28,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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+2T+p5T2 1 + 2 T + p^{5} T^{2}
5 1+96T+p5T2 1 + 96 T + p^{5} T^{2}
11 1+720T+p5T2 1 + 720 T + p^{5} T^{2}
13 1−44pT+p5T2 1 - 44 p T + p^{5} T^{2}
17 1−1254T+p5T2 1 - 1254 T + p^{5} T^{2}
19 1+94T+p5T2 1 + 94 T + p^{5} T^{2}
23 1−96T+p5T2 1 - 96 T + p^{5} T^{2}
29 1+4374T+p5T2 1 + 4374 T + p^{5} T^{2}
31 1+6244T+p5T2 1 + 6244 T + p^{5} T^{2}
37 1+10798T+p5T2 1 + 10798 T + p^{5} T^{2}
41 1−12006T+p5T2 1 - 12006 T + p^{5} T^{2}
43 1+9160T+p5T2 1 + 9160 T + p^{5} T^{2}
47 1+25836T+p5T2 1 + 25836 T + p^{5} T^{2}
53 1−1014T+p5T2 1 - 1014 T + p^{5} T^{2}
59 1−1242T+p5T2 1 - 1242 T + p^{5} T^{2}
61 1−7592T+p5T2 1 - 7592 T + p^{5} T^{2}
67 1−41132T+p5T2 1 - 41132 T + p^{5} T^{2}
71 1+37632T+p5T2 1 + 37632 T + p^{5} T^{2}
73 1+13438T+p5T2 1 + 13438 T + p^{5} T^{2}
79 1−6248T+p5T2 1 - 6248 T + p^{5} T^{2}
83 1+25254T+p5T2 1 + 25254 T + p^{5} T^{2}
89 1+45126T+p5T2 1 + 45126 T + p^{5} T^{2}
97 1−107222T+p5T2 1 - 107222 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

−15.66460879568356725976406438718, −14.56727787606432360931467156002, −12.89076813217567077613700172951, −11.58112929535376601244632812409, −10.75986348281136906360165818499, −8.434082554525968076475551344576, −7.62447871447066040952538120986, −5.31965875775595161984552193376, −3.40240128430237200615070717741, 0, 3.40240128430237200615070717741, 5.31965875775595161984552193376, 7.62447871447066040952538120986, 8.434082554525968076475551344576, 10.75986348281136906360165818499, 11.58112929535376601244632812409, 12.89076813217567077613700172951, 14.56727787606432360931467156002, 15.66460879568356725976406438718

Graph of the ZZ-function along the critical line