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Properties

Label 2-1440-1.1-c3-0-3
Degree 22
Conductor 14401440
Sign 11
Analytic cond. 84.962784.9627
Root an. cond. 9.217529.21752
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  − 5·5-s − 4·7-s − 40·11-s − 90·13-s + 70·17-s + 40·19-s − 108·23-s + 25·25-s − 166·29-s − 40·31-s + 20·35-s − 130·37-s + 310·41-s − 268·43-s + 556·47-s − 327·49-s + 370·53-s + 200·55-s − 240·59-s − 130·61-s + 450·65-s + 876·67-s + 840·71-s + 250·73-s + 160·77-s − 880·79-s + 188·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.215·7-s − 1.09·11-s − 1.92·13-s + 0.998·17-s + 0.482·19-s − 0.979·23-s + 1/5·25-s − 1.06·29-s − 0.231·31-s + 0.0965·35-s − 0.577·37-s + 1.18·41-s − 0.950·43-s + 1.72·47-s − 0.953·49-s + 0.958·53-s + 0.490·55-s − 0.529·59-s − 0.272·61-s + 0.858·65-s + 1.59·67-s + 1.40·71-s + 0.400·73-s + 0.236·77-s − 1.25·79-s + 0.248·83-s + ⋯

Functional equation

Λ(s)=(1440s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1440s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14401440    =    25⋅32⋅52^{5} \cdot 3^{2} \cdot 5
Sign: 11
Analytic conductor: 84.962784.9627
Root analytic conductor: 9.217529.21752
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1440, ( :3/2), 1)(2,\ 1440,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.94551405070.9455140507
L(12)L(\frac12) ≈\approx 0.94551405070.9455140507
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
5 1+pT 1 + p T
good7 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
11 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
13 1+90T+p3T2 1 + 90 T + p^{3} T^{2}
17 1−70T+p3T2 1 - 70 T + p^{3} T^{2}
19 1−40T+p3T2 1 - 40 T + p^{3} T^{2}
23 1+108T+p3T2 1 + 108 T + p^{3} T^{2}
29 1+166T+p3T2 1 + 166 T + p^{3} T^{2}
31 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
37 1+130T+p3T2 1 + 130 T + p^{3} T^{2}
41 1−310T+p3T2 1 - 310 T + p^{3} T^{2}
43 1+268T+p3T2 1 + 268 T + p^{3} T^{2}
47 1−556T+p3T2 1 - 556 T + p^{3} T^{2}
53 1−370T+p3T2 1 - 370 T + p^{3} T^{2}
59 1+240T+p3T2 1 + 240 T + p^{3} T^{2}
61 1+130T+p3T2 1 + 130 T + p^{3} T^{2}
67 1−876T+p3T2 1 - 876 T + p^{3} T^{2}
71 1−840T+p3T2 1 - 840 T + p^{3} T^{2}
73 1−250T+p3T2 1 - 250 T + p^{3} T^{2}
79 1+880T+p3T2 1 + 880 T + p^{3} T^{2}
83 1−188T+p3T2 1 - 188 T + p^{3} T^{2}
89 1−726T+p3T2 1 - 726 T + p^{3} T^{2}
97 1+1550T+p3T2 1 + 1550 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

−9.355927509627335852473984556907, −8.086919121630046029862346378856, −7.62668597849544585483371635958, −6.96692062986203239792854121155, −5.62248139232690598008883731919, −5.12646340000089761802342101097, −4.03089947384815292088439117493, −2.99388485233309469256197650230, −2.10796208857406477375355135679, −0.45362168349166410274509632703, 0.45362168349166410274509632703, 2.10796208857406477375355135679, 2.99388485233309469256197650230, 4.03089947384815292088439117493, 5.12646340000089761802342101097, 5.62248139232690598008883731919, 6.96692062986203239792854121155, 7.62668597849544585483371635958, 8.086919121630046029862346378856, 9.355927509627335852473984556907

Graph of the ZZ-function along the critical line