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Properties

Label 2-30e2-1.1-c3-0-21
Degree 22
Conductor 900900
Sign −1-1
Analytic cond. 53.101753.1017
Root an. cond. 7.287097.28709
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 26·7-s − 45·11-s + 44·13-s − 117·17-s − 91·19-s + 18·23-s − 144·29-s + 26·31-s − 214·37-s + 459·41-s − 460·43-s + 468·47-s + 333·49-s − 558·53-s + 72·59-s − 118·61-s + 251·67-s − 108·71-s + 299·73-s − 1.17e3·77-s − 898·79-s − 927·83-s − 351·89-s + 1.14e3·91-s + 386·97-s + 954·101-s − 772·103-s + ⋯
L(s)  = 1  + 1.40·7-s − 1.23·11-s + 0.938·13-s − 1.66·17-s − 1.09·19-s + 0.163·23-s − 0.922·29-s + 0.150·31-s − 0.950·37-s + 1.74·41-s − 1.63·43-s + 1.45·47-s + 0.970·49-s − 1.44·53-s + 0.158·59-s − 0.247·61-s + 0.457·67-s − 0.180·71-s + 0.479·73-s − 1.73·77-s − 1.27·79-s − 1.22·83-s − 0.418·89-s + 1.31·91-s + 0.404·97-s + 0.939·101-s − 0.738·103-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 53.101753.1017
Root analytic conductor: 7.287097.28709
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 900, ( :3/2), −1)(2,\ 900,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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 1
good7 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
11 1+45T+p3T2 1 + 45 T + p^{3} T^{2}
13 1−44T+p3T2 1 - 44 T + p^{3} T^{2}
17 1+117T+p3T2 1 + 117 T + p^{3} T^{2}
19 1+91T+p3T2 1 + 91 T + p^{3} T^{2}
23 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
29 1+144T+p3T2 1 + 144 T + p^{3} T^{2}
31 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
37 1+214T+p3T2 1 + 214 T + p^{3} T^{2}
41 1−459T+p3T2 1 - 459 T + p^{3} T^{2}
43 1+460T+p3T2 1 + 460 T + p^{3} T^{2}
47 1−468T+p3T2 1 - 468 T + p^{3} T^{2}
53 1+558T+p3T2 1 + 558 T + p^{3} T^{2}
59 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
61 1+118T+p3T2 1 + 118 T + p^{3} T^{2}
67 1−251T+p3T2 1 - 251 T + p^{3} T^{2}
71 1+108T+p3T2 1 + 108 T + p^{3} T^{2}
73 1−299T+p3T2 1 - 299 T + p^{3} T^{2}
79 1+898T+p3T2 1 + 898 T + p^{3} T^{2}
83 1+927T+p3T2 1 + 927 T + p^{3} T^{2}
89 1+351T+p3T2 1 + 351 T + p^{3} T^{2}
97 1−386T+p3T2 1 - 386 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

−9.067097859850360137490642193981, −8.435301088174339859196192379885, −7.78564118842372946995127301916, −6.75990995819829741589509138519, −5.71262902248282020416523314000, −4.81616111022271634887205969740, −4.04755402334840306342624864315, −2.52257418223036102878174608543, −1.62131451557060714015625674328, 0, 1.62131451557060714015625674328, 2.52257418223036102878174608543, 4.04755402334840306342624864315, 4.81616111022271634887205969740, 5.71262902248282020416523314000, 6.75990995819829741589509138519, 7.78564118842372946995127301916, 8.435301088174339859196192379885, 9.067097859850360137490642193981

Graph of the ZZ-function along the critical line