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Properties

Label 2-33e2-1.1-c3-0-63
Degree 22
Conductor 10891089
Sign −1-1
Analytic cond. 64.253064.2530
Root an. cond. 8.015808.01580
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  − 8·4-s − 18·5-s + 64·16-s + 144·20-s + 108·23-s + 199·25-s + 340·31-s − 434·37-s + 36·47-s − 343·49-s + 738·53-s + 720·59-s − 512·64-s − 416·67-s − 612·71-s − 1.15e3·80-s − 1.67e3·89-s − 864·92-s − 34·97-s − 1.59e3·100-s + 1.17e3·103-s − 2.14e3·113-s − 1.94e3·115-s + ⋯
L(s)  = 1  − 4-s − 1.60·5-s + 16-s + 1.60·20-s + 0.979·23-s + 1.59·25-s + 1.96·31-s − 1.92·37-s + 0.111·47-s − 49-s + 1.91·53-s + 1.58·59-s − 64-s − 0.758·67-s − 1.02·71-s − 1.60·80-s − 1.99·89-s − 0.979·92-s − 0.0355·97-s − 1.59·100-s + 1.12·103-s − 1.78·113-s − 1.57·115-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 64.253064.2530
Root analytic conductor: 8.015808.01580
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1089, ( :3/2), −1)(2,\ 1089,\ (\ :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)
bad3 1 1
11 1 1
good2 1+p3T2 1 + p^{3} T^{2}
5 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
7 1+p3T2 1 + p^{3} T^{2}
13 1+p3T2 1 + p^{3} T^{2}
17 1+p3T2 1 + p^{3} T^{2}
19 1+p3T2 1 + p^{3} T^{2}
23 1−108T+p3T2 1 - 108 T + p^{3} T^{2}
29 1+p3T2 1 + p^{3} T^{2}
31 1−340T+p3T2 1 - 340 T + p^{3} T^{2}
37 1+434T+p3T2 1 + 434 T + p^{3} T^{2}
41 1+p3T2 1 + p^{3} T^{2}
43 1+p3T2 1 + p^{3} T^{2}
47 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
53 1−738T+p3T2 1 - 738 T + p^{3} T^{2}
59 1−720T+p3T2 1 - 720 T + p^{3} T^{2}
61 1+p3T2 1 + p^{3} T^{2}
67 1+416T+p3T2 1 + 416 T + p^{3} T^{2}
71 1+612T+p3T2 1 + 612 T + p^{3} T^{2}
73 1+p3T2 1 + p^{3} T^{2}
79 1+p3T2 1 + p^{3} T^{2}
83 1+p3T2 1 + p^{3} T^{2}
89 1+1674T+p3T2 1 + 1674 T + p^{3} T^{2}
97 1+34T+p3T2 1 + 34 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

−8.736785677830695573996941291180, −8.446377138974293221095131063340, −7.53715153686217732863227816235, −6.75574963478171206685547455048, −5.37322498552637436041865688150, −4.56837298844257880324597562311, −3.85429552969987359634357864643, −2.98709261059238423344664646162, −1.00767604415401823796887979437, 0, 1.00767604415401823796887979437, 2.98709261059238423344664646162, 3.85429552969987359634357864643, 4.56837298844257880324597562311, 5.37322498552637436041865688150, 6.75574963478171206685547455048, 7.53715153686217732863227816235, 8.446377138974293221095131063340, 8.736785677830695573996941291180

Graph of the ZZ-function along the critical line