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Properties

Label 2-693-1.1-c3-0-14
Degree 22
Conductor 693693
Sign 11
Analytic cond. 40.888340.8883
Root an. cond. 6.394396.39439
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  − 4.93·2-s + 16.3·4-s + 14.7·5-s − 7·7-s − 41.1·8-s − 72.8·10-s − 11·11-s − 32.3·13-s + 34.5·14-s + 72.4·16-s − 45.2·17-s − 146.·19-s + 241.·20-s + 54.2·22-s + 153.·23-s + 93.2·25-s + 159.·26-s − 114.·28-s + 78.6·29-s + 106.·31-s − 27.8·32-s + 223.·34-s − 103.·35-s + 94.0·37-s + 723.·38-s − 608.·40-s + 417.·41-s + ⋯
L(s)  = 1  − 1.74·2-s + 2.04·4-s + 1.32·5-s − 0.377·7-s − 1.81·8-s − 2.30·10-s − 0.301·11-s − 0.689·13-s + 0.659·14-s + 1.13·16-s − 0.645·17-s − 1.76·19-s + 2.69·20-s + 0.525·22-s + 1.38·23-s + 0.746·25-s + 1.20·26-s − 0.772·28-s + 0.503·29-s + 0.614·31-s − 0.153·32-s + 1.12·34-s − 0.499·35-s + 0.418·37-s + 3.08·38-s − 2.40·40-s + 1.58·41-s + ⋯

Functional equation

Λ(s)=(693s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(693s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 693693    =    32⋅7⋅113^{2} \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 40.888340.8883
Root analytic conductor: 6.394396.39439
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 693, ( :3/2), 1)(2,\ 693,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.87652094110.8765209411
L(12)L(\frac12) ≈\approx 0.87652094110.8765209411
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)
bad3 1 1
7 1+7T 1 + 7T
11 1+11T 1 + 11T
good2 1+4.93T+8T2 1 + 4.93T + 8T^{2}
5 1−14.7T+125T2 1 - 14.7T + 125T^{2}
13 1+32.3T+2.19e3T2 1 + 32.3T + 2.19e3T^{2}
17 1+45.2T+4.91e3T2 1 + 45.2T + 4.91e3T^{2}
19 1+146.T+6.85e3T2 1 + 146.T + 6.85e3T^{2}
23 1−153.T+1.21e4T2 1 - 153.T + 1.21e4T^{2}
29 1−78.6T+2.43e4T2 1 - 78.6T + 2.43e4T^{2}
31 1−106.T+2.97e4T2 1 - 106.T + 2.97e4T^{2}
37 1−94.0T+5.06e4T2 1 - 94.0T + 5.06e4T^{2}
41 1−417.T+6.89e4T2 1 - 417.T + 6.89e4T^{2}
43 1−60.3T+7.95e4T2 1 - 60.3T + 7.95e4T^{2}
47 1−253.T+1.03e5T2 1 - 253.T + 1.03e5T^{2}
53 1+647.T+1.48e5T2 1 + 647.T + 1.48e5T^{2}
59 1−559.T+2.05e5T2 1 - 559.T + 2.05e5T^{2}
61 1+602.T+2.26e5T2 1 + 602.T + 2.26e5T^{2}
67 1−343.T+3.00e5T2 1 - 343.T + 3.00e5T^{2}
71 1+224.T+3.57e5T2 1 + 224.T + 3.57e5T^{2}
73 1−1.05e3T+3.89e5T2 1 - 1.05e3T + 3.89e5T^{2}
79 1+102.T+4.93e5T2 1 + 102.T + 4.93e5T^{2}
83 1−730.T+5.71e5T2 1 - 730.T + 5.71e5T^{2}
89 1+43.7T+7.04e5T2 1 + 43.7T + 7.04e5T^{2}
97 1−8.01T+9.12e5T2 1 - 8.01T + 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

−9.900629192511521821974283451247, −9.243017763076846509153279089879, −8.658423506224411195911725841008, −7.60866628890741955982625660137, −6.63443061736748684489987972802, −6.10344484070771777809950231673, −4.70889538151091165132304053734, −2.66799663689542625372205408401, −2.04751301985597671675548059527, −0.68353817373044458304890663313, 0.68353817373044458304890663313, 2.04751301985597671675548059527, 2.66799663689542625372205408401, 4.70889538151091165132304053734, 6.10344484070771777809950231673, 6.63443061736748684489987972802, 7.60866628890741955982625660137, 8.658423506224411195911725841008, 9.243017763076846509153279089879, 9.900629192511521821974283451247

Graph of the ZZ-function along the critical line