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Properties

Label 2-798-1.1-c3-0-23
Degree 22
Conductor 798798
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s + 4·4-s + 21.5·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s − 43.0·10-s − 33.5·11-s + 12·12-s − 46.6·13-s − 14·14-s + 64.5·15-s + 16·16-s − 41.8·17-s − 18·18-s − 19·19-s + 86.1·20-s + 21·21-s + 67.0·22-s + 52.5·23-s − 24·24-s + 338.·25-s + 93.3·26-s + 27·27-s + 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.92·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.36·10-s − 0.918·11-s + 0.288·12-s − 0.996·13-s − 0.267·14-s + 1.11·15-s + 0.250·16-s − 0.597·17-s − 0.235·18-s − 0.229·19-s + 0.962·20-s + 0.218·21-s + 0.649·22-s + 0.476·23-s − 0.204·24-s + 2.70·25-s + 0.704·26-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: 11
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: 00
Selberg data: (2, 798, ( :3/2), 1)(2,\ 798,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.6847264762.684726476
L(12)L(\frac12) ≈\approx 2.6847264762.684726476
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 1−21.5T+125T2 1 - 21.5T + 125T^{2}
11 1+33.5T+1.33e3T2 1 + 33.5T + 1.33e3T^{2}
13 1+46.6T+2.19e3T2 1 + 46.6T + 2.19e3T^{2}
17 1+41.8T+4.91e3T2 1 + 41.8T + 4.91e3T^{2}
23 1−52.5T+1.21e4T2 1 - 52.5T + 1.21e4T^{2}
29 1−198.T+2.43e4T2 1 - 198.T + 2.43e4T^{2}
31 1−117.T+2.97e4T2 1 - 117.T + 2.97e4T^{2}
37 1−240.T+5.06e4T2 1 - 240.T + 5.06e4T^{2}
41 1+3.25T+6.89e4T2 1 + 3.25T + 6.89e4T^{2}
43 1−486.T+7.95e4T2 1 - 486.T + 7.95e4T^{2}
47 1+176.T+1.03e5T2 1 + 176.T + 1.03e5T^{2}
53 1−751.T+1.48e5T2 1 - 751.T + 1.48e5T^{2}
59 1−306.T+2.05e5T2 1 - 306.T + 2.05e5T^{2}
61 1−39.2T+2.26e5T2 1 - 39.2T + 2.26e5T^{2}
67 1−582.T+3.00e5T2 1 - 582.T + 3.00e5T^{2}
71 1−586.T+3.57e5T2 1 - 586.T + 3.57e5T^{2}
73 1+759.T+3.89e5T2 1 + 759.T + 3.89e5T^{2}
79 1+300.T+4.93e5T2 1 + 300.T + 4.93e5T^{2}
83 1+826.T+5.71e5T2 1 + 826.T + 5.71e5T^{2}
89 1−455.T+7.04e5T2 1 - 455.T + 7.04e5T^{2}
97 1+528.T+9.12e5T2 1 + 528.T + 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

−9.962291906364829906743264331424, −9.074229844172003532598177775645, −8.418597743550267119083914220460, −7.36302323682049974800606716779, −6.50887901282638452657954649451, −5.53688647781536747230479761819, −4.65372271397177601728398297227, −2.55247184454947843764340915107, −2.40577868878851121014792129784, −1.02734491058775931517426532333, 1.02734491058775931517426532333, 2.40577868878851121014792129784, 2.55247184454947843764340915107, 4.65372271397177601728398297227, 5.53688647781536747230479761819, 6.50887901282638452657954649451, 7.36302323682049974800606716779, 8.418597743550267119083914220460, 9.074229844172003532598177775645, 9.962291906364829906743264331424

Graph of the ZZ-function along the critical line