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Properties

Label 2-882-1.1-c3-0-13
Degree 22
Conductor 882882
Sign 11
Analytic cond. 52.039652.0396
Root an. cond. 7.213857.21385
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 + 4·4-s − 3.44·5-s + 8·8-s − 6.89·10-s − 36.1·11-s − 10.2·13-s + 16·16-s + 118.·17-s + 38.6·19-s − 13.7·20-s − 72.2·22-s + 36.2·23-s − 113.·25-s − 20.4·26-s + 12.1·29-s + 145.·31-s + 32·32-s + 236.·34-s − 1.37·37-s + 77.3·38-s − 27.5·40-s + 168·41-s + 299.·43-s − 144.·44-s + 72.4·46-s + 502.·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.308·5-s + 0.353·8-s − 0.217·10-s − 0.990·11-s − 0.218·13-s + 0.250·16-s + 1.69·17-s + 0.466·19-s − 0.154·20-s − 0.700·22-s + 0.328·23-s − 0.904·25-s − 0.154·26-s + 0.0776·29-s + 0.842·31-s + 0.176·32-s + 1.19·34-s − 0.00609·37-s + 0.330·38-s − 0.108·40-s + 0.639·41-s + 1.06·43-s − 0.495·44-s + 0.232·46-s + 1.56·47-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 52.039652.0396
Root analytic conductor: 7.213857.21385
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 882, ( :3/2), 1)(2,\ 882,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.0624002583.062400258
L(12)L(\frac12) ≈\approx 3.0624002583.062400258
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 1
7 1 1
good5 1+3.44T+125T2 1 + 3.44T + 125T^{2}
11 1+36.1T+1.33e3T2 1 + 36.1T + 1.33e3T^{2}
13 1+10.2T+2.19e3T2 1 + 10.2T + 2.19e3T^{2}
17 1−118.T+4.91e3T2 1 - 118.T + 4.91e3T^{2}
19 1−38.6T+6.85e3T2 1 - 38.6T + 6.85e3T^{2}
23 1−36.2T+1.21e4T2 1 - 36.2T + 1.21e4T^{2}
29 1−12.1T+2.43e4T2 1 - 12.1T + 2.43e4T^{2}
31 1−145.T+2.97e4T2 1 - 145.T + 2.97e4T^{2}
37 1+1.37T+5.06e4T2 1 + 1.37T + 5.06e4T^{2}
41 1−168T+6.89e4T2 1 - 168T + 6.89e4T^{2}
43 1−299.T+7.95e4T2 1 - 299.T + 7.95e4T^{2}
47 1−502.T+1.03e5T2 1 - 502.T + 1.03e5T^{2}
53 1−625.T+1.48e5T2 1 - 625.T + 1.48e5T^{2}
59 1+42.2T+2.05e5T2 1 + 42.2T + 2.05e5T^{2}
61 1−439.T+2.26e5T2 1 - 439.T + 2.26e5T^{2}
67 1+763.T+3.00e5T2 1 + 763.T + 3.00e5T^{2}
71 1+1.02e3T+3.57e5T2 1 + 1.02e3T + 3.57e5T^{2}
73 1+579.T+3.89e5T2 1 + 579.T + 3.89e5T^{2}
79 1−942.T+4.93e5T2 1 - 942.T + 4.93e5T^{2}
83 1+474.T+5.71e5T2 1 + 474.T + 5.71e5T^{2}
89 1−821.T+7.04e5T2 1 - 821.T + 7.04e5T^{2}
97 1−1.10e3T+9.12e5T2 1 - 1.10e3T + 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

−10.03803170006646590271638951988, −8.857692341334340253098475684309, −7.63881009103077876018362556518, −7.46811562653675112526241842432, −5.98703332250907106742164165093, −5.40642832280840336912521422771, −4.38375003381521360112231310915, −3.34908533914813252848959114073, −2.43574305680165991560157288464, −0.873397623437876555672511769782, 0.873397623437876555672511769782, 2.43574305680165991560157288464, 3.34908533914813252848959114073, 4.38375003381521360112231310915, 5.40642832280840336912521422771, 5.98703332250907106742164165093, 7.46811562653675112526241842432, 7.63881009103077876018362556518, 8.857692341334340253098475684309, 10.03803170006646590271638951988

Graph of the ZZ-function along the critical line