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Properties

Label 2-153-1.1-c3-0-6
Degree 22
Conductor 153153
Sign 11
Analytic cond. 9.027299.02729
Root an. cond. 3.004543.00454
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  + 2-s − 7·4-s + 20·5-s − 2·7-s − 15·8-s + 20·10-s + 48·11-s − 14·13-s − 2·14-s + 41·16-s + 17·17-s + 92·19-s − 140·20-s + 48·22-s + 122·23-s + 275·25-s − 14·26-s + 14·28-s + 36·29-s − 182·31-s + 161·32-s + 17·34-s − 40·35-s + 76·37-s + 92·38-s − 300·40-s − 294·41-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s + 1.78·5-s − 0.107·7-s − 0.662·8-s + 0.632·10-s + 1.31·11-s − 0.298·13-s − 0.0381·14-s + 0.640·16-s + 0.242·17-s + 1.11·19-s − 1.56·20-s + 0.465·22-s + 1.10·23-s + 11/5·25-s − 0.105·26-s + 0.0944·28-s + 0.230·29-s − 1.05·31-s + 0.889·32-s + 0.0857·34-s − 0.193·35-s + 0.337·37-s + 0.392·38-s − 1.18·40-s − 1.11·41-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: 11
Analytic conductor: 9.027299.02729
Root analytic conductor: 3.004543.00454
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 153, ( :3/2), 1)(2,\ 153,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.2613519852.261351985
L(12)L(\frac12) ≈\approx 2.2613519852.261351985
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
17 1−pT 1 - p T
good2 1−T+p3T2 1 - T + p^{3} T^{2}
5 1−4pT+p3T2 1 - 4 p T + p^{3} T^{2}
7 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
11 1−48T+p3T2 1 - 48 T + p^{3} T^{2}
13 1+14T+p3T2 1 + 14 T + p^{3} T^{2}
19 1−92T+p3T2 1 - 92 T + p^{3} T^{2}
23 1−122T+p3T2 1 - 122 T + p^{3} T^{2}
29 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
31 1+182T+p3T2 1 + 182 T + p^{3} T^{2}
37 1−76T+p3T2 1 - 76 T + p^{3} T^{2}
41 1+294T+p3T2 1 + 294 T + p^{3} T^{2}
43 1+428T+p3T2 1 + 428 T + p^{3} T^{2}
47 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
53 1−234T+p3T2 1 - 234 T + p^{3} T^{2}
59 1−540T+p3T2 1 - 540 T + p^{3} T^{2}
61 1+820T+p3T2 1 + 820 T + p^{3} T^{2}
67 1−700T+p3T2 1 - 700 T + p^{3} T^{2}
71 1+794T+p3T2 1 + 794 T + p^{3} T^{2}
73 1+1038T+p3T2 1 + 1038 T + p^{3} T^{2}
79 1−858T+p3T2 1 - 858 T + p^{3} T^{2}
83 1+1052T+p3T2 1 + 1052 T + p^{3} T^{2}
89 1+1102T+p3T2 1 + 1102 T + p^{3} T^{2}
97 1−710T+p3T2 1 - 710 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

−12.84572032586095150129070280011, −11.68873954698695547292313537973, −10.12340851965310588877862971314, −9.468567662220988120960864940459, −8.777328738222094048189913970631, −6.87966353094760050551239872917, −5.78427121583277909087922347724, −4.90786977862438319421197028568, −3.24550987121121250197287421243, −1.38427781376150785635910606005, 1.38427781376150785635910606005, 3.24550987121121250197287421243, 4.90786977862438319421197028568, 5.78427121583277909087922347724, 6.87966353094760050551239872917, 8.777328738222094048189913970631, 9.468567662220988120960864940459, 10.12340851965310588877862971314, 11.68873954698695547292313537973, 12.84572032586095150129070280011

Graph of the ZZ-function along the critical line