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Properties

Label 2-1440-1.1-c3-0-7
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 − 30·7-s + 50·11-s − 88·13-s − 74·17-s − 140·19-s + 80·23-s + 25·25-s + 234·29-s − 150·35-s + 116·37-s + 72·41-s − 280·43-s + 120·47-s + 557·49-s + 498·53-s + 250·55-s − 870·59-s + 650·61-s − 440·65-s − 420·67-s + 1.02e3·71-s − 322·73-s − 1.50e3·77-s − 160·79-s − 980·83-s − 370·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.61·7-s + 1.37·11-s − 1.87·13-s − 1.05·17-s − 1.69·19-s + 0.725·23-s + 1/5·25-s + 1.49·29-s − 0.724·35-s + 0.515·37-s + 0.274·41-s − 0.993·43-s + 0.372·47-s + 1.62·49-s + 1.29·53-s + 0.612·55-s − 1.91·59-s + 1.36·61-s − 0.839·65-s − 0.765·67-s + 1.70·71-s − 0.516·73-s − 2.22·77-s − 0.227·79-s − 1.29·83-s − 0.472·85-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 1.2695978981.269597898
L(12)L(\frac12) ≈\approx 1.2695978981.269597898
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+30T+p3T2 1 + 30 T + p^{3} T^{2}
11 1−50T+p3T2 1 - 50 T + p^{3} T^{2}
13 1+88T+p3T2 1 + 88 T + p^{3} T^{2}
17 1+74T+p3T2 1 + 74 T + p^{3} T^{2}
19 1+140T+p3T2 1 + 140 T + p^{3} T^{2}
23 1−80T+p3T2 1 - 80 T + p^{3} T^{2}
29 1−234T+p3T2 1 - 234 T + p^{3} T^{2}
31 1+p3T2 1 + p^{3} T^{2}
37 1−116T+p3T2 1 - 116 T + p^{3} T^{2}
41 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
43 1+280T+p3T2 1 + 280 T + p^{3} T^{2}
47 1−120T+p3T2 1 - 120 T + p^{3} T^{2}
53 1−498T+p3T2 1 - 498 T + p^{3} T^{2}
59 1+870T+p3T2 1 + 870 T + p^{3} T^{2}
61 1−650T+p3T2 1 - 650 T + p^{3} T^{2}
67 1+420T+p3T2 1 + 420 T + p^{3} T^{2}
71 1−1020T+p3T2 1 - 1020 T + p^{3} T^{2}
73 1+322T+p3T2 1 + 322 T + p^{3} T^{2}
79 1+160T+p3T2 1 + 160 T + p^{3} T^{2}
83 1+980T+p3T2 1 + 980 T + p^{3} T^{2}
89 1−1124T+p3T2 1 - 1124 T + p^{3} T^{2}
97 1−1114T+p3T2 1 - 1114 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

−9.170770449175621346641164037392, −8.698067835935893108881640921367, −7.25145577137105647403766833281, −6.56085985836188356127498568580, −6.25861228117705552377472002338, −4.86388977716025109580838227857, −4.09493565939268740159186022999, −2.91591103789219758503069008077, −2.14227206263947767241181183518, −0.52728645845441789911851267592, 0.52728645845441789911851267592, 2.14227206263947767241181183518, 2.91591103789219758503069008077, 4.09493565939268740159186022999, 4.86388977716025109580838227857, 6.25861228117705552377472002338, 6.56085985836188356127498568580, 7.25145577137105647403766833281, 8.698067835935893108881640921367, 9.170770449175621346641164037392

Graph of the ZZ-function along the critical line