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Properties

Label 2-1872-1.1-c1-0-20
Degree 22
Conductor 18721872
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 13-s − 2·17-s + 4·19-s − 25-s − 6·29-s − 2·37-s − 6·41-s + 12·43-s − 4·47-s − 7·49-s − 6·53-s − 8·59-s − 2·61-s − 2·65-s − 4·67-s − 12·71-s − 14·73-s + 8·83-s + 4·85-s + 18·89-s − 8·95-s − 6·97-s − 14·101-s − 16·103-s + 4·107-s − 2·109-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.277·13-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 1.11·29-s − 0.328·37-s − 0.937·41-s + 1.82·43-s − 0.583·47-s − 49-s − 0.824·53-s − 1.04·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s − 1.42·71-s − 1.63·73-s + 0.878·83-s + 0.433·85-s + 1.90·89-s − 0.820·95-s − 0.609·97-s − 1.39·101-s − 1.57·103-s + 0.386·107-s − 0.191·109-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: −1-1
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: 11
Selberg data: (2, 1872, ( :1/2), −1)(2,\ 1872,\ (\ :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
13 1−T 1 - T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+4T+pT2 1 + 4 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+pT2 1 + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1−18T+pT2 1 - 18 T + p T^{2}
97 1+6T+pT2 1 + 6 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

−8.879911485018581552559778228194, −7.87497595233942443821507658391, −7.48206171863010651464390955453, −6.50109862386502065678161037272, −5.60510042774675865346276845379, −4.63629406156287515269954623878, −3.80747723245115179033242709882, −2.97565069723020365742526516584, −1.57297786911914117520864059523, 0, 1.57297786911914117520864059523, 2.97565069723020365742526516584, 3.80747723245115179033242709882, 4.63629406156287515269954623878, 5.60510042774675865346276845379, 6.50109862386502065678161037272, 7.48206171863010651464390955453, 7.87497595233942443821507658391, 8.879911485018581552559778228194

Graph of the ZZ-function along the critical line