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Properties

Label 2-847-1.1-c3-0-9
Degree 22
Conductor 847847
Sign 11
Analytic cond. 49.974649.9746
Root an. cond. 7.069277.06927
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  + 3.24·2-s − 5.49·3-s + 2.53·4-s − 16.0·5-s − 17.8·6-s + 7·7-s − 17.7·8-s + 3.16·9-s − 52.2·10-s − 13.9·12-s − 35.3·13-s + 22.7·14-s + 88.4·15-s − 77.8·16-s − 40.4·17-s + 10.2·18-s − 118.·19-s − 40.7·20-s − 38.4·21-s − 174.·23-s + 97.4·24-s + 134.·25-s − 114.·26-s + 130.·27-s + 17.7·28-s + 262.·29-s + 286.·30-s + ⋯
L(s)  = 1  + 1.14·2-s − 1.05·3-s + 0.316·4-s − 1.43·5-s − 1.21·6-s + 0.377·7-s − 0.784·8-s + 0.117·9-s − 1.65·10-s − 0.334·12-s − 0.754·13-s + 0.433·14-s + 1.52·15-s − 1.21·16-s − 0.577·17-s + 0.134·18-s − 1.42·19-s − 0.455·20-s − 0.399·21-s − 1.58·23-s + 0.828·24-s + 1.07·25-s − 0.865·26-s + 0.933·27-s + 0.119·28-s + 1.68·29-s + 1.74·30-s + ⋯

Functional equation

Λ(s)=(847s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(847s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 847847    =    7⋅1127 \cdot 11^{2}
Sign: 11
Analytic conductor: 49.974649.9746
Root analytic conductor: 7.069277.06927
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 847, ( :3/2), 1)(2,\ 847,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.51661957260.5166195726
L(12)L(\frac12) ≈\approx 0.51661957260.5166195726
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)
bad7 1−7T 1 - 7T
11 1 1
good2 1−3.24T+8T2 1 - 3.24T + 8T^{2}
3 1+5.49T+27T2 1 + 5.49T + 27T^{2}
5 1+16.0T+125T2 1 + 16.0T + 125T^{2}
13 1+35.3T+2.19e3T2 1 + 35.3T + 2.19e3T^{2}
17 1+40.4T+4.91e3T2 1 + 40.4T + 4.91e3T^{2}
19 1+118.T+6.85e3T2 1 + 118.T + 6.85e3T^{2}
23 1+174.T+1.21e4T2 1 + 174.T + 1.21e4T^{2}
29 1−262.T+2.43e4T2 1 - 262.T + 2.43e4T^{2}
31 1+36.1T+2.97e4T2 1 + 36.1T + 2.97e4T^{2}
37 1−19.0T+5.06e4T2 1 - 19.0T + 5.06e4T^{2}
41 1+156.T+6.89e4T2 1 + 156.T + 6.89e4T^{2}
43 1+287.T+7.95e4T2 1 + 287.T + 7.95e4T^{2}
47 1−397.T+1.03e5T2 1 - 397.T + 1.03e5T^{2}
53 1−272.T+1.48e5T2 1 - 272.T + 1.48e5T^{2}
59 1+507.T+2.05e5T2 1 + 507.T + 2.05e5T^{2}
61 1+35.5T+2.26e5T2 1 + 35.5T + 2.26e5T^{2}
67 1−979.T+3.00e5T2 1 - 979.T + 3.00e5T^{2}
71 1−750.T+3.57e5T2 1 - 750.T + 3.57e5T^{2}
73 1+395.T+3.89e5T2 1 + 395.T + 3.89e5T^{2}
79 1−736.T+4.93e5T2 1 - 736.T + 4.93e5T^{2}
83 1+582.T+5.71e5T2 1 + 582.T + 5.71e5T^{2}
89 1+806.T+7.04e5T2 1 + 806.T + 7.04e5T^{2}
97 1+957.T+9.12e5T2 1 + 957.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

−10.15364617340704496479691036112, −8.692165156886550043103518769685, −8.086766936372657198370434050822, −6.85499734329122203101798381548, −6.19087026861756053298857249678, −5.11961573762048942432391398474, −4.47912252751872200064768012615, −3.83635299478462737059996578924, −2.48715025033123340933215350948, −0.33210805545570359202129519817, 0.33210805545570359202129519817, 2.48715025033123340933215350948, 3.83635299478462737059996578924, 4.47912252751872200064768012615, 5.11961573762048942432391398474, 6.19087026861756053298857249678, 6.85499734329122203101798381548, 8.086766936372657198370434050822, 8.692165156886550043103518769685, 10.15364617340704496479691036112

Graph of the ZZ-function along the critical line