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Properties

Label 2-798-1.1-c3-0-32
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 + 4.39·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s + 8.78·10-s + 50.1·11-s + 12·12-s + 11.4·13-s + 14·14-s + 13.1·15-s + 16·16-s + 107.·17-s + 18·18-s + 19·19-s + 17.5·20-s + 21·21-s + 100.·22-s − 181.·23-s + 24·24-s − 105.·25-s + 22.9·26-s + 27·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.392·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.277·10-s + 1.37·11-s + 0.288·12-s + 0.244·13-s + 0.267·14-s + 0.226·15-s + 0.250·16-s + 1.53·17-s + 0.235·18-s + 0.229·19-s + 0.196·20-s + 0.218·21-s + 0.971·22-s − 1.64·23-s + 0.204·24-s − 0.845·25-s + 0.173·26-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 5.1901884695.190188469
L(12)L(\frac12) ≈\approx 5.1901884695.190188469
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 1−4.39T+125T2 1 - 4.39T + 125T^{2}
11 1−50.1T+1.33e3T2 1 - 50.1T + 1.33e3T^{2}
13 1−11.4T+2.19e3T2 1 - 11.4T + 2.19e3T^{2}
17 1−107.T+4.91e3T2 1 - 107.T + 4.91e3T^{2}
23 1+181.T+1.21e4T2 1 + 181.T + 1.21e4T^{2}
29 1+286.T+2.43e4T2 1 + 286.T + 2.43e4T^{2}
31 1+85.4T+2.97e4T2 1 + 85.4T + 2.97e4T^{2}
37 1−313.T+5.06e4T2 1 - 313.T + 5.06e4T^{2}
41 1+18.8T+6.89e4T2 1 + 18.8T + 6.89e4T^{2}
43 1−395.T+7.95e4T2 1 - 395.T + 7.95e4T^{2}
47 1−364.T+1.03e5T2 1 - 364.T + 1.03e5T^{2}
53 1−152.T+1.48e5T2 1 - 152.T + 1.48e5T^{2}
59 1+375.T+2.05e5T2 1 + 375.T + 2.05e5T^{2}
61 1−253.T+2.26e5T2 1 - 253.T + 2.26e5T^{2}
67 1+291.T+3.00e5T2 1 + 291.T + 3.00e5T^{2}
71 1−587.T+3.57e5T2 1 - 587.T + 3.57e5T^{2}
73 1+788.T+3.89e5T2 1 + 788.T + 3.89e5T^{2}
79 1−929.T+4.93e5T2 1 - 929.T + 4.93e5T^{2}
83 1+371.T+5.71e5T2 1 + 371.T + 5.71e5T^{2}
89 1−914.T+7.04e5T2 1 - 914.T + 7.04e5T^{2}
97 1−374.T+9.12e5T2 1 - 374.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

−9.721626286440941666967277283521, −9.218843065289097000880638221309, −7.949716217421617354939913963149, −7.40360098163101451066690947611, −6.10513715377299416031223415152, −5.60079944768666754458588170464, −4.12527041213543073867454797320, −3.64643314807451000373069265319, −2.22758824285708646146733114899, −1.27935284882845820672065244683, 1.27935284882845820672065244683, 2.22758824285708646146733114899, 3.64643314807451000373069265319, 4.12527041213543073867454797320, 5.60079944768666754458588170464, 6.10513715377299416031223415152, 7.40360098163101451066690947611, 7.949716217421617354939913963149, 9.218843065289097000880638221309, 9.721626286440941666967277283521

Graph of the ZZ-function along the critical line