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Properties

Label 2-28-1.1-c3-0-1
Degree 22
Conductor 2828
Sign −1-1
Analytic cond. 1.652051.65205
Root an. cond. 1.285321.28532
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·3-s − 8·5-s − 7·7-s + 73·9-s − 40·11-s − 12·13-s + 80·15-s − 58·17-s + 26·19-s + 70·21-s − 64·23-s − 61·25-s − 460·27-s − 62·29-s + 252·31-s + 400·33-s + 56·35-s + 26·37-s + 120·39-s + 6·41-s + 416·43-s − 584·45-s − 396·47-s + 49·49-s + 580·51-s − 450·53-s + 320·55-s + ⋯
L(s)  = 1  − 1.92·3-s − 0.715·5-s − 0.377·7-s + 2.70·9-s − 1.09·11-s − 0.256·13-s + 1.37·15-s − 0.827·17-s + 0.313·19-s + 0.727·21-s − 0.580·23-s − 0.487·25-s − 3.27·27-s − 0.397·29-s + 1.46·31-s + 2.11·33-s + 0.270·35-s + 0.115·37-s + 0.492·39-s + 0.0228·41-s + 1.47·43-s − 1.93·45-s − 1.22·47-s + 1/7·49-s + 1.59·51-s − 1.16·53-s + 0.784·55-s + ⋯

Functional equation

Λ(s)=(28s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(28s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2828    =    22⋅72^{2} \cdot 7
Sign: −1-1
Analytic conductor: 1.652051.65205
Root analytic conductor: 1.285321.28532
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 28, ( :3/2), −1)(2,\ 28,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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+pT 1 + p T
good3 1+10T+p3T2 1 + 10 T + p^{3} T^{2}
5 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
11 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
13 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
17 1+58T+p3T2 1 + 58 T + p^{3} T^{2}
19 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
23 1+64T+p3T2 1 + 64 T + p^{3} T^{2}
29 1+62T+p3T2 1 + 62 T + p^{3} T^{2}
31 1−252T+p3T2 1 - 252 T + p^{3} T^{2}
37 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
41 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
43 1−416T+p3T2 1 - 416 T + p^{3} T^{2}
47 1+396T+p3T2 1 + 396 T + p^{3} T^{2}
53 1+450T+p3T2 1 + 450 T + p^{3} T^{2}
59 1−274T+p3T2 1 - 274 T + p^{3} T^{2}
61 1+576T+p3T2 1 + 576 T + p^{3} T^{2}
67 1+476T+p3T2 1 + 476 T + p^{3} T^{2}
71 1+448T+p3T2 1 + 448 T + p^{3} T^{2}
73 1+158T+p3T2 1 + 158 T + p^{3} T^{2}
79 1+936T+p3T2 1 + 936 T + p^{3} T^{2}
83 1−530T+p3T2 1 - 530 T + p^{3} T^{2}
89 1+390T+p3T2 1 + 390 T + p^{3} T^{2}
97 1−214T+p3T2 1 - 214 T + p^{3} 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

−16.07092329469791305387008900168, −15.58857037284035452257426767028, −13.17493072879754927160972315449, −12.10544057288673767820092826412, −11.14271482983331810332690506926, −10.02979999855798524146877493937, −7.53741591071156022841236447136, −6.07979688505861171575806799387, −4.59778668307750420844259673078, 0, 4.59778668307750420844259673078, 6.07979688505861171575806799387, 7.53741591071156022841236447136, 10.02979999855798524146877493937, 11.14271482983331810332690506926, 12.10544057288673767820092826412, 13.17493072879754927160972315449, 15.58857037284035452257426767028, 16.07092329469791305387008900168

Graph of the ZZ-function along the critical line