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Properties

Label 2-30e2-1.1-c3-0-15
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  − 13·7-s − 6·11-s + 5·13-s + 78·17-s + 65·19-s − 138·23-s − 66·29-s + 299·31-s − 214·37-s − 360·41-s + 203·43-s − 78·47-s − 174·49-s − 636·53-s − 786·59-s + 467·61-s − 217·67-s + 360·71-s − 286·73-s + 78·77-s + 272·79-s − 498·83-s − 65·91-s − 511·97-s + 1.81e3·101-s − 1.70e3·103-s − 1.23e3·107-s + ⋯
L(s)  = 1  − 0.701·7-s − 0.164·11-s + 0.106·13-s + 1.11·17-s + 0.784·19-s − 1.25·23-s − 0.422·29-s + 1.73·31-s − 0.950·37-s − 1.37·41-s + 0.719·43-s − 0.242·47-s − 0.507·49-s − 1.64·53-s − 1.73·59-s + 0.980·61-s − 0.395·67-s + 0.601·71-s − 0.458·73-s + 0.115·77-s + 0.387·79-s − 0.658·83-s − 0.0748·91-s − 0.534·97-s + 1.78·101-s − 1.63·103-s − 1.11·107-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+13T+p3T2 1 + 13 T + p^{3} T^{2}
11 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
13 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
17 1−78T+p3T2 1 - 78 T + p^{3} T^{2}
19 1−65T+p3T2 1 - 65 T + p^{3} T^{2}
23 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
29 1+66T+p3T2 1 + 66 T + p^{3} T^{2}
31 1−299T+p3T2 1 - 299 T + p^{3} T^{2}
37 1+214T+p3T2 1 + 214 T + p^{3} T^{2}
41 1+360T+p3T2 1 + 360 T + p^{3} T^{2}
43 1−203T+p3T2 1 - 203 T + p^{3} T^{2}
47 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
53 1+12pT+p3T2 1 + 12 p T + p^{3} T^{2}
59 1+786T+p3T2 1 + 786 T + p^{3} T^{2}
61 1−467T+p3T2 1 - 467 T + p^{3} T^{2}
67 1+217T+p3T2 1 + 217 T + p^{3} T^{2}
71 1−360T+p3T2 1 - 360 T + p^{3} T^{2}
73 1+286T+p3T2 1 + 286 T + p^{3} T^{2}
79 1−272T+p3T2 1 - 272 T + p^{3} T^{2}
83 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
89 1+p3T2 1 + p^{3} T^{2}
97 1+511T+p3T2 1 + 511 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

−9.569606235179620032877442203128, −8.353626751179776151582937505479, −7.69917994605502400072877346541, −6.65342552078132698315102450649, −5.87581161801070877355223699777, −4.91970529534829414223163678505, −3.68986477595632227870410713524, −2.88384454312624688425462428218, −1.43026964004570665953934602828, 0, 1.43026964004570665953934602828, 2.88384454312624688425462428218, 3.68986477595632227870410713524, 4.91970529534829414223163678505, 5.87581161801070877355223699777, 6.65342552078132698315102450649, 7.69917994605502400072877346541, 8.353626751179776151582937505479, 9.569606235179620032877442203128

Graph of the ZZ-function along the critical line