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Properties

Label 2-798-1.1-c3-0-48
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 + 11.9·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s − 23.8·10-s − 9.57·11-s + 12·12-s − 59.5·13-s + 14·14-s + 35.7·15-s + 16·16-s + 52.8·17-s − 18·18-s − 19·19-s + 47.7·20-s − 21·21-s + 19.1·22-s − 103.·23-s − 24·24-s + 17.3·25-s + 119.·26-s + 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.06·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.754·10-s − 0.262·11-s + 0.288·12-s − 1.26·13-s + 0.267·14-s + 0.616·15-s + 0.250·16-s + 0.753·17-s − 0.235·18-s − 0.229·19-s + 0.533·20-s − 0.218·21-s + 0.185·22-s − 0.939·23-s − 0.204·24-s + 0.138·25-s + 0.897·26-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 1−11.9T+125T2 1 - 11.9T + 125T^{2}
11 1+9.57T+1.33e3T2 1 + 9.57T + 1.33e3T^{2}
13 1+59.5T+2.19e3T2 1 + 59.5T + 2.19e3T^{2}
17 1−52.8T+4.91e3T2 1 - 52.8T + 4.91e3T^{2}
23 1+103.T+1.21e4T2 1 + 103.T + 1.21e4T^{2}
29 1+182.T+2.43e4T2 1 + 182.T + 2.43e4T^{2}
31 1+99.0T+2.97e4T2 1 + 99.0T + 2.97e4T^{2}
37 1+301.T+5.06e4T2 1 + 301.T + 5.06e4T^{2}
41 1+100.T+6.89e4T2 1 + 100.T + 6.89e4T^{2}
43 1+140.T+7.95e4T2 1 + 140.T + 7.95e4T^{2}
47 1−234.T+1.03e5T2 1 - 234.T + 1.03e5T^{2}
53 1−253.T+1.48e5T2 1 - 253.T + 1.48e5T^{2}
59 1+410.T+2.05e5T2 1 + 410.T + 2.05e5T^{2}
61 1+250.T+2.26e5T2 1 + 250.T + 2.26e5T^{2}
67 1−272.T+3.00e5T2 1 - 272.T + 3.00e5T^{2}
71 1+332.T+3.57e5T2 1 + 332.T + 3.57e5T^{2}
73 1−544.T+3.89e5T2 1 - 544.T + 3.89e5T^{2}
79 1−879.T+4.93e5T2 1 - 879.T + 4.93e5T^{2}
83 1+200.T+5.71e5T2 1 + 200.T + 5.71e5T^{2}
89 1+954.T+7.04e5T2 1 + 954.T + 7.04e5T^{2}
97 1+1.39e3T+9.12e5T2 1 + 1.39e3T + 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

−9.612029698837025473311530061349, −8.773158267228061085172086766732, −7.75737096618784202256028070463, −7.07965234174733616014271917819, −6.00059661493869955474218710313, −5.16707838113381127767239126593, −3.63110788759666411519177057018, −2.46376941903874804906698979457, −1.71521295673885339944884725488, 0, 1.71521295673885339944884725488, 2.46376941903874804906698979457, 3.63110788759666411519177057018, 5.16707838113381127767239126593, 6.00059661493869955474218710313, 7.07965234174733616014271917819, 7.75737096618784202256028070463, 8.773158267228061085172086766732, 9.612029698837025473311530061349

Graph of the ZZ-function along the critical line