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Properties

Label 2-1152-1.1-c1-0-17
Degree 22
Conductor 11521152
Sign −1-1
Analytic cond. 9.198769.19876
Root an. cond. 3.032943.03294
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 4·7-s − 2·11-s − 2·13-s + 2·17-s − 2·19-s − 4·23-s − 25-s − 6·29-s − 8·35-s − 10·37-s + 6·41-s − 6·43-s + 8·47-s + 9·49-s − 6·53-s − 4·55-s + 14·59-s − 2·61-s − 4·65-s − 10·67-s − 12·71-s + 14·73-s + 8·77-s − 8·79-s − 6·83-s + 4·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.51·7-s − 0.603·11-s − 0.554·13-s + 0.485·17-s − 0.458·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s − 1.35·35-s − 1.64·37-s + 0.937·41-s − 0.914·43-s + 1.16·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s + 1.82·59-s − 0.256·61-s − 0.496·65-s − 1.22·67-s − 1.42·71-s + 1.63·73-s + 0.911·77-s − 0.900·79-s − 0.658·83-s + 0.433·85-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: −1-1
Analytic conductor: 9.198769.19876
Root analytic conductor: 3.032943.03294
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1152, ( :1/2), −1)(2,\ 1152,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
good5 1−2T+pT2 1 - 2 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−14T+pT2 1 - 14 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1+2T+pT2 1 + 2 T + p 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

−9.604101000606496287963353743481, −8.759776534720553396196404795081, −7.62957697295837367057177448880, −6.80509880453900742773724680138, −5.94298026933117547539540451748, −5.37965797861855942151102168788, −3.98566652809305192679300882554, −2.97266024838274409285785943283, −1.98277685475006180754048515240, 0, 1.98277685475006180754048515240, 2.97266024838274409285785943283, 3.98566652809305192679300882554, 5.37965797861855942151102168788, 5.94298026933117547539540451748, 6.80509880453900742773724680138, 7.62957697295837367057177448880, 8.759776534720553396196404795081, 9.604101000606496287963353743481

Graph of the ZZ-function along the critical line