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Properties

Label 2-99-1.1-c3-0-8
Degree 22
Conductor 9999
Sign 11
Analytic cond. 5.841185.84118
Root an. cond. 2.416852.41685
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 17·4-s + 14·5-s − 32·7-s + 45·8-s + 70·10-s + 11·11-s − 38·13-s − 160·14-s + 89·16-s + 2·17-s + 72·19-s + 238·20-s + 55·22-s − 68·23-s + 71·25-s − 190·26-s − 544·28-s + 54·29-s − 152·31-s + 85·32-s + 10·34-s − 448·35-s + 174·37-s + 360·38-s + 630·40-s − 94·41-s + ⋯
L(s)  = 1  + 1.76·2-s + 17/8·4-s + 1.25·5-s − 1.72·7-s + 1.98·8-s + 2.21·10-s + 0.301·11-s − 0.810·13-s − 3.05·14-s + 1.39·16-s + 0.0285·17-s + 0.869·19-s + 2.66·20-s + 0.533·22-s − 0.616·23-s + 0.567·25-s − 1.43·26-s − 3.67·28-s + 0.345·29-s − 0.880·31-s + 0.469·32-s + 0.0504·34-s − 2.16·35-s + 0.773·37-s + 1.53·38-s + 2.49·40-s − 0.358·41-s + ⋯

Functional equation

Λ(s)=(99s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(99s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9999    =    32⋅113^{2} \cdot 11
Sign: 11
Analytic conductor: 5.841185.84118
Root analytic conductor: 2.416852.41685
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 99, ( :3/2), 1)(2,\ 99,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 4.1187402574.118740257
L(12)L(\frac12) ≈\approx 4.1187402574.118740257
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
11 1−pT 1 - p T
good2 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
5 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
7 1+32T+p3T2 1 + 32 T + p^{3} T^{2}
13 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
17 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
19 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
23 1+68T+p3T2 1 + 68 T + p^{3} T^{2}
29 1−54T+p3T2 1 - 54 T + p^{3} T^{2}
31 1+152T+p3T2 1 + 152 T + p^{3} T^{2}
37 1−174T+p3T2 1 - 174 T + p^{3} T^{2}
41 1+94T+p3T2 1 + 94 T + p^{3} T^{2}
43 1+528T+p3T2 1 + 528 T + p^{3} T^{2}
47 1−340T+p3T2 1 - 340 T + p^{3} T^{2}
53 1−438T+p3T2 1 - 438 T + p^{3} T^{2}
59 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
61 1−570T+p3T2 1 - 570 T + p^{3} T^{2}
67 1+460T+p3T2 1 + 460 T + p^{3} T^{2}
71 1−1092T+p3T2 1 - 1092 T + p^{3} T^{2}
73 1−562T+p3T2 1 - 562 T + p^{3} T^{2}
79 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
83 1+372T+p3T2 1 + 372 T + p^{3} T^{2}
89 1−966T+p3T2 1 - 966 T + p^{3} T^{2}
97 1+526T+p3T2 1 + 526 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

−13.43434910137316612141022374741, −12.71125030925909594058041984741, −11.83235099854909654277750922131, −10.21423808108031227871944916599, −9.454680857221027075949373686511, −7.02366367103417308769025902453, −6.18528858126133271259285952533, −5.29499404212907390026734482434, −3.60958730236608261490424709415, −2.40684343637818605205952601774, 2.40684343637818605205952601774, 3.60958730236608261490424709415, 5.29499404212907390026734482434, 6.18528858126133271259285952533, 7.02366367103417308769025902453, 9.454680857221027075949373686511, 10.21423808108031227871944916599, 11.83235099854909654277750922131, 12.71125030925909594058041984741, 13.43434910137316612141022374741

Graph of the ZZ-function along the critical line