Properties

Label 2-2280-1.1-c1-0-5
Degree 22
Conductor 22802280
Sign 11
Analytic cond. 18.205818.2058
Root an. cond. 4.266834.26683
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·7-s + 9-s − 4·11-s − 2·13-s − 15-s + 19-s − 2·21-s + 8·23-s + 25-s + 27-s − 2·29-s + 10·31-s − 4·33-s + 2·35-s + 6·37-s − 2·39-s + 12·41-s + 8·43-s − 45-s + 8·47-s − 3·49-s − 6·53-s + 4·55-s + 57-s − 6·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.229·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.79·31-s − 0.696·33-s + 0.338·35-s + 0.986·37-s − 0.320·39-s + 1.87·41-s + 1.21·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.539·55-s + 0.132·57-s − 0.781·59-s + ⋯

Functional equation

Λ(s)=(2280s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2280s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22802280    =    2335192^{3} \cdot 3 \cdot 5 \cdot 19
Sign: 11
Analytic conductor: 18.205818.2058
Root analytic conductor: 4.266834.26683
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2280, ( :1/2), 1)(2,\ 2280,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 1.6383188921.638318892
L(12)L(\frac12) \approx 1.6383188921.638318892
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1T 1 - T
5 1+T 1 + T
19 1T 1 - T
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
23 18T+pT2 1 - 8 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 110T+pT2 1 - 10 T + p T^{2}
37 16T+pT2 1 - 6 T + p T^{2}
41 112T+pT2 1 - 12 T + p T^{2}
43 18T+pT2 1 - 8 T + p T^{2}
47 18T+pT2 1 - 8 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 110T+pT2 1 - 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 110T+pT2 1 - 10 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+2T+pT2 1 + 2 T + p T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.147419761271871272354642937619, −8.094892290976825000288656742302, −7.60454808441980285120122662578, −6.87026565439754025500762199807, −5.89915736099352673808280918566, −4.92416488810909051901283951074, −4.14212641887134945016435736327, −2.92936483576798041344267272221, −2.61725081168138791694652154838, −0.792182304489764548174574988163, 0.792182304489764548174574988163, 2.61725081168138791694652154838, 2.92936483576798041344267272221, 4.14212641887134945016435736327, 4.92416488810909051901283951074, 5.89915736099352673808280918566, 6.87026565439754025500762199807, 7.60454808441980285120122662578, 8.094892290976825000288656742302, 9.147419761271871272354642937619

Graph of the ZZ-function along the critical line