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Properties

Label 2-30e2-1.1-c3-0-13
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  − 22·7-s + 14·11-s + 30·13-s + 62·17-s − 120·19-s + 188·23-s − 96·29-s + 184·31-s − 406·37-s − 130·41-s − 148·43-s + 448·47-s + 141·49-s − 414·53-s − 266·59-s − 838·61-s − 248·67-s − 1.02e3·71-s − 484·73-s − 308·77-s − 48·79-s + 548·83-s + 650·89-s − 660·91-s + 1.81e3·97-s − 1.68e3·101-s + 298·103-s + ⋯
L(s)  = 1  − 1.18·7-s + 0.383·11-s + 0.640·13-s + 0.884·17-s − 1.44·19-s + 1.70·23-s − 0.614·29-s + 1.06·31-s − 1.80·37-s − 0.495·41-s − 0.524·43-s + 1.39·47-s + 0.411·49-s − 1.07·53-s − 0.586·59-s − 1.75·61-s − 0.452·67-s − 1.70·71-s − 0.775·73-s − 0.455·77-s − 0.0683·79-s + 0.724·83-s + 0.774·89-s − 0.760·91-s + 1.90·97-s − 1.66·101-s + 0.285·103-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+22T+p3T2 1 + 22 T + p^{3} T^{2}
11 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
13 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
17 1−62T+p3T2 1 - 62 T + p^{3} T^{2}
19 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
23 1−188T+p3T2 1 - 188 T + p^{3} T^{2}
29 1+96T+p3T2 1 + 96 T + p^{3} T^{2}
31 1−184T+p3T2 1 - 184 T + p^{3} T^{2}
37 1+406T+p3T2 1 + 406 T + p^{3} T^{2}
41 1+130T+p3T2 1 + 130 T + p^{3} T^{2}
43 1+148T+p3T2 1 + 148 T + p^{3} T^{2}
47 1−448T+p3T2 1 - 448 T + p^{3} T^{2}
53 1+414T+p3T2 1 + 414 T + p^{3} T^{2}
59 1+266T+p3T2 1 + 266 T + p^{3} T^{2}
61 1+838T+p3T2 1 + 838 T + p^{3} T^{2}
67 1+248T+p3T2 1 + 248 T + p^{3} T^{2}
71 1+1020T+p3T2 1 + 1020 T + p^{3} T^{2}
73 1+484T+p3T2 1 + 484 T + p^{3} T^{2}
79 1+48T+p3T2 1 + 48 T + p^{3} T^{2}
83 1−548T+p3T2 1 - 548 T + p^{3} T^{2}
89 1−650T+p3T2 1 - 650 T + p^{3} T^{2}
97 1−1816T+p3T2 1 - 1816 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

−9.161785831462419252722957916942, −8.719532153645374580350524102807, −7.51010471249783789909434307221, −6.60064635526440518878699333779, −6.03217374992231065185497025269, −4.84010864228896918404101802234, −3.67358707023879935614062915546, −2.93227359806216301692346059997, −1.40524800629792220099746244878, 0, 1.40524800629792220099746244878, 2.93227359806216301692346059997, 3.67358707023879935614062915546, 4.84010864228896918404101802234, 6.03217374992231065185497025269, 6.60064635526440518878699333779, 7.51010471249783789909434307221, 8.719532153645374580350524102807, 9.161785831462419252722957916942

Graph of the ZZ-function along the critical line