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Properties

Label 2-882-1.1-c3-0-34
Degree 22
Conductor 882882
Sign −1-1
Analytic cond. 52.039652.0396
Root an. cond. 7.213857.21385
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 6·5-s − 8·8-s − 12·10-s − 30·11-s − 2·13-s + 16·16-s + 66·17-s + 52·19-s + 24·20-s + 60·22-s − 114·23-s − 89·25-s + 4·26-s − 72·29-s + 196·31-s − 32·32-s − 132·34-s − 286·37-s − 104·38-s − 48·40-s − 378·41-s + 164·43-s − 120·44-s + 228·46-s − 228·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.536·5-s − 0.353·8-s − 0.379·10-s − 0.822·11-s − 0.0426·13-s + 1/4·16-s + 0.941·17-s + 0.627·19-s + 0.268·20-s + 0.581·22-s − 1.03·23-s − 0.711·25-s + 0.0301·26-s − 0.461·29-s + 1.13·31-s − 0.176·32-s − 0.665·34-s − 1.27·37-s − 0.443·38-s − 0.189·40-s − 1.43·41-s + 0.581·43-s − 0.411·44-s + 0.730·46-s − 0.707·47-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: −1-1
Analytic conductor: 52.039652.0396
Root analytic conductor: 7.213857.21385
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 882, ( :3/2), −1)(2,\ 882,\ (\ :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+pT 1 + p T
3 1 1
7 1 1
good5 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
11 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
13 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
17 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
19 1−52T+p3T2 1 - 52 T + p^{3} T^{2}
23 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
29 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
31 1−196T+p3T2 1 - 196 T + p^{3} T^{2}
37 1+286T+p3T2 1 + 286 T + p^{3} T^{2}
41 1+378T+p3T2 1 + 378 T + p^{3} T^{2}
43 1−164T+p3T2 1 - 164 T + p^{3} T^{2}
47 1+228T+p3T2 1 + 228 T + p^{3} T^{2}
53 1−348T+p3T2 1 - 348 T + p^{3} T^{2}
59 1+348T+p3T2 1 + 348 T + p^{3} T^{2}
61 1−106T+p3T2 1 - 106 T + p^{3} T^{2}
67 1−596T+p3T2 1 - 596 T + p^{3} T^{2}
71 1+630T+p3T2 1 + 630 T + p^{3} T^{2}
73 1−1042T+p3T2 1 - 1042 T + p^{3} T^{2}
79 1+88T+p3T2 1 + 88 T + p^{3} T^{2}
83 1+1440T+p3T2 1 + 1440 T + p^{3} T^{2}
89 1−1374T+p3T2 1 - 1374 T + p^{3} T^{2}
97 1−34T+p3T2 1 - 34 T + p^{3} T^{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

−9.558998757719835334369672290920, −8.383214222939481173119503532025, −7.82639343617231311496367007621, −6.87414883245019214785990900827, −5.84607362542699487640131407267, −5.15000225408950394502137665612, −3.64332908176095657160530089066, −2.51601187284871032215428564529, −1.43666607632335274257302355314, 0, 1.43666607632335274257302355314, 2.51601187284871032215428564529, 3.64332908176095657160530089066, 5.15000225408950394502137665612, 5.84607362542699487640131407267, 6.87414883245019214785990900827, 7.82639343617231311496367007621, 8.383214222939481173119503532025, 9.558998757719835334369672290920

Graph of the ZZ-function along the critical line