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Properties

Label 2-432-1.1-c3-0-11
Degree 22
Conductor 432432
Sign 11
Analytic cond. 25.488825.4888
Root an. cond. 5.048645.04864
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  + 15·5-s + 25·7-s + 15·11-s + 20·13-s + 72·17-s − 2·19-s − 114·23-s + 100·25-s + 30·29-s − 101·31-s + 375·35-s − 430·37-s − 30·41-s − 110·43-s + 330·47-s + 282·49-s + 621·53-s + 225·55-s + 660·59-s − 376·61-s + 300·65-s + 250·67-s + 360·71-s + 785·73-s + 375·77-s − 488·79-s − 489·83-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.34·7-s + 0.411·11-s + 0.426·13-s + 1.02·17-s − 0.0241·19-s − 1.03·23-s + 4/5·25-s + 0.192·29-s − 0.585·31-s + 1.81·35-s − 1.91·37-s − 0.114·41-s − 0.390·43-s + 1.02·47-s + 0.822·49-s + 1.60·53-s + 0.551·55-s + 1.45·59-s − 0.789·61-s + 0.572·65-s + 0.455·67-s + 0.601·71-s + 1.25·73-s + 0.555·77-s − 0.694·79-s − 0.646·83-s + ⋯

Functional equation

Λ(s)=(432s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(432s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 432432    =    24⋅332^{4} \cdot 3^{3}
Sign: 11
Analytic conductor: 25.488825.4888
Root analytic conductor: 5.048645.04864
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 432, ( :3/2), 1)(2,\ 432,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.0503721573.050372157
L(12)L(\frac12) ≈\approx 3.0503721573.050372157
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 1
3 1 1
good5 1−3pT+p3T2 1 - 3 p T + p^{3} T^{2}
7 1−25T+p3T2 1 - 25 T + p^{3} T^{2}
11 1−15T+p3T2 1 - 15 T + p^{3} T^{2}
13 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
17 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
19 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
23 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
29 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
31 1+101T+p3T2 1 + 101 T + p^{3} T^{2}
37 1+430T+p3T2 1 + 430 T + p^{3} T^{2}
41 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
43 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
47 1−330T+p3T2 1 - 330 T + p^{3} T^{2}
53 1−621T+p3T2 1 - 621 T + p^{3} T^{2}
59 1−660T+p3T2 1 - 660 T + p^{3} T^{2}
61 1+376T+p3T2 1 + 376 T + p^{3} T^{2}
67 1−250T+p3T2 1 - 250 T + p^{3} T^{2}
71 1−360T+p3T2 1 - 360 T + p^{3} T^{2}
73 1−785T+p3T2 1 - 785 T + p^{3} T^{2}
79 1+488T+p3T2 1 + 488 T + p^{3} T^{2}
83 1+489T+p3T2 1 + 489 T + p^{3} T^{2}
89 1+450T+p3T2 1 + 450 T + p^{3} T^{2}
97 1+1105T+p3T2 1 + 1105 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

−10.59199335777222656212673297287, −9.944550218558164504304461614812, −8.904790318448454321747239782596, −8.115474735284305216473207260849, −6.96686166741679146077798305058, −5.78352649374207826925743295618, −5.19845343951551946234578824278, −3.82763111411129667701989521839, −2.15762877241039560527557218277, −1.30551836432768445183321594073, 1.30551836432768445183321594073, 2.15762877241039560527557218277, 3.82763111411129667701989521839, 5.19845343951551946234578824278, 5.78352649374207826925743295618, 6.96686166741679146077798305058, 8.115474735284305216473207260849, 8.904790318448454321747239782596, 9.944550218558164504304461614812, 10.59199335777222656212673297287

Graph of the ZZ-function along the critical line