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Properties

Label 2-1872-1.1-c1-0-1
Degree 22
Conductor 18721872
Sign 11
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
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  − 2·5-s − 4·7-s − 4·11-s + 13-s − 2·17-s + 8·19-s − 25-s − 6·29-s + 4·31-s + 8·35-s − 2·37-s + 10·41-s − 4·43-s + 8·47-s + 9·49-s + 10·53-s + 8·55-s + 4·59-s − 2·61-s − 2·65-s + 16·67-s − 8·71-s + 2·73-s + 16·77-s − 8·79-s + 12·83-s + 4·85-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 1.20·11-s + 0.277·13-s − 0.485·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 1.35·35-s − 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s + 1.07·55-s + 0.520·59-s − 0.256·61-s − 0.248·65-s + 1.95·67-s − 0.949·71-s + 0.234·73-s + 1.82·77-s − 0.900·79-s + 1.31·83-s + 0.433·85-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1872, ( :1/2), 1)(2,\ 1872,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.83740957240.8374095724
L(12)L(\frac12) ≈\approx 0.83740957240.8374095724
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
13 1−T 1 - T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−9.345209736202926164014285850515, −8.395061304144126538425647052449, −7.52465381698855157679013184836, −7.08365389517628564377574744262, −5.97887006638506534112113716589, −5.29861144478199351873889204714, −4.06906259281866598276648207762, −3.34937914407959923888310141579, −2.53219499761269661738776398133, −0.58989713961815991805077071088, 0.58989713961815991805077071088, 2.53219499761269661738776398133, 3.34937914407959923888310141579, 4.06906259281866598276648207762, 5.29861144478199351873889204714, 5.97887006638506534112113716589, 7.08365389517628564377574744262, 7.52465381698855157679013184836, 8.395061304144126538425647052449, 9.345209736202926164014285850515

Graph of the ZZ-function along the critical line