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Properties

Label 2-798-1.1-c3-0-49
Degree 22
Conductor 798798
Sign −1-1
Analytic cond. 47.083547.0835
Root an. cond. 6.861746.86174
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s + 4·4-s + 5.41·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 10.8·10-s + 17.0·11-s + 12·12-s − 49.6·13-s − 14·14-s + 16.2·15-s + 16·16-s − 110.·17-s − 18·18-s + 19·19-s + 21.6·20-s + 21·21-s − 34.1·22-s − 30.9·23-s − 24·24-s − 95.6·25-s + 99.3·26-s + 27·27-s + 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.484·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.342·10-s + 0.467·11-s + 0.288·12-s − 1.05·13-s − 0.267·14-s + 0.279·15-s + 0.250·16-s − 1.58·17-s − 0.235·18-s + 0.229·19-s + 0.242·20-s + 0.218·21-s − 0.330·22-s − 0.280·23-s − 0.204·24-s − 0.765·25-s + 0.749·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(798s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(798s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 798798    =    2⋅3⋅7⋅192 \cdot 3 \cdot 7 \cdot 19
Sign: −1-1
Analytic conductor: 47.083547.0835
Root analytic conductor: 6.861746.86174
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 798, ( :3/2), −1)(2,\ 798,\ (\ :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+2T 1 + 2T
3 1−3T 1 - 3T
7 1−7T 1 - 7T
19 1−19T 1 - 19T
good5 1−5.41T+125T2 1 - 5.41T + 125T^{2}
11 1−17.0T+1.33e3T2 1 - 17.0T + 1.33e3T^{2}
13 1+49.6T+2.19e3T2 1 + 49.6T + 2.19e3T^{2}
17 1+110.T+4.91e3T2 1 + 110.T + 4.91e3T^{2}
23 1+30.9T+1.21e4T2 1 + 30.9T + 1.21e4T^{2}
29 1+172.T+2.43e4T2 1 + 172.T + 2.43e4T^{2}
31 1+267.T+2.97e4T2 1 + 267.T + 2.97e4T^{2}
37 1−299.T+5.06e4T2 1 - 299.T + 5.06e4T^{2}
41 1−108.T+6.89e4T2 1 - 108.T + 6.89e4T^{2}
43 1+322.T+7.95e4T2 1 + 322.T + 7.95e4T^{2}
47 1+426.T+1.03e5T2 1 + 426.T + 1.03e5T^{2}
53 1+136.T+1.48e5T2 1 + 136.T + 1.48e5T^{2}
59 1−303.T+2.05e5T2 1 - 303.T + 2.05e5T^{2}
61 1−419.T+2.26e5T2 1 - 419.T + 2.26e5T^{2}
67 1+709.T+3.00e5T2 1 + 709.T + 3.00e5T^{2}
71 1−37.6T+3.57e5T2 1 - 37.6T + 3.57e5T^{2}
73 1−836.T+3.89e5T2 1 - 836.T + 3.89e5T^{2}
79 1+52.3T+4.93e5T2 1 + 52.3T + 4.93e5T^{2}
83 1+762.T+5.71e5T2 1 + 762.T + 5.71e5T^{2}
89 1+406.T+7.04e5T2 1 + 406.T + 7.04e5T^{2}
97 1−1.48e3T+9.12e5T2 1 - 1.48e3T + 9.12e5T^{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

−9.442339699112908368546813392859, −8.756449709106958247689353917212, −7.79037787494325622105102366005, −7.08989259955434359846419116113, −6.12162998910597024856959408668, −4.94561268003871861868810050121, −3.80026894394417091995783385595, −2.39215471760742314962668481036, −1.71767966597873652647834231975, 0, 1.71767966597873652647834231975, 2.39215471760742314962668481036, 3.80026894394417091995783385595, 4.94561268003871861868810050121, 6.12162998910597024856959408668, 7.08989259955434359846419116113, 7.79037787494325622105102366005, 8.756449709106958247689353917212, 9.442339699112908368546813392859

Graph of the ZZ-function along the critical line