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Properties

Label 2-798-1.1-c3-0-22
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 − 16.2·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 32.5·10-s + 29.7·11-s − 12·12-s − 46.5·13-s + 14·14-s + 48.8·15-s + 16·16-s − 12.5·17-s − 18·18-s + 19·19-s − 65.1·20-s + 21·21-s − 59.5·22-s + 113.·23-s + 24·24-s + 140.·25-s + 93.1·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.45·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.03·10-s + 0.816·11-s − 0.288·12-s − 0.993·13-s + 0.267·14-s + 0.841·15-s + 0.250·16-s − 0.178·17-s − 0.235·18-s + 0.229·19-s − 0.728·20-s + 0.218·21-s − 0.577·22-s + 1.02·23-s + 0.204·24-s + 1.12·25-s + 0.702·26-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+16.2T+125T2 1 + 16.2T + 125T^{2}
11 1−29.7T+1.33e3T2 1 - 29.7T + 1.33e3T^{2}
13 1+46.5T+2.19e3T2 1 + 46.5T + 2.19e3T^{2}
17 1+12.5T+4.91e3T2 1 + 12.5T + 4.91e3T^{2}
23 1−113.T+1.21e4T2 1 - 113.T + 1.21e4T^{2}
29 1−156.T+2.43e4T2 1 - 156.T + 2.43e4T^{2}
31 1+97.1T+2.97e4T2 1 + 97.1T + 2.97e4T^{2}
37 1−230.T+5.06e4T2 1 - 230.T + 5.06e4T^{2}
41 1−204.T+6.89e4T2 1 - 204.T + 6.89e4T^{2}
43 1+41.5T+7.95e4T2 1 + 41.5T + 7.95e4T^{2}
47 1+26.4T+1.03e5T2 1 + 26.4T + 1.03e5T^{2}
53 1−465.T+1.48e5T2 1 - 465.T + 1.48e5T^{2}
59 1+52.9T+2.05e5T2 1 + 52.9T + 2.05e5T^{2}
61 1+32.4T+2.26e5T2 1 + 32.4T + 2.26e5T^{2}
67 1+568.T+3.00e5T2 1 + 568.T + 3.00e5T^{2}
71 1+376.T+3.57e5T2 1 + 376.T + 3.57e5T^{2}
73 1+1.03e3T+3.89e5T2 1 + 1.03e3T + 3.89e5T^{2}
79 1−452.T+4.93e5T2 1 - 452.T + 4.93e5T^{2}
83 1+222.T+5.71e5T2 1 + 222.T + 5.71e5T^{2}
89 1−765.T+7.04e5T2 1 - 765.T + 7.04e5T^{2}
97 1−105.T+9.12e5T2 1 - 105.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

−9.405694502414496120383874223338, −8.655318567592197616504934040307, −7.56287205090650517403840845193, −7.12689816361981885260474888891, −6.17882936044691037959103450225, −4.85283898169297330079317730664, −3.94358173359349410867211565202, −2.78758337654086005500480969957, −1.02446314920538743099175162483, 0, 1.02446314920538743099175162483, 2.78758337654086005500480969957, 3.94358173359349410867211565202, 4.85283898169297330079317730664, 6.17882936044691037959103450225, 7.12689816361981885260474888891, 7.56287205090650517403840845193, 8.655318567592197616504934040307, 9.405694502414496120383874223338

Graph of the ZZ-function along the critical line