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Properties

Label 2-8280-1.1-c1-0-86
Degree 22
Conductor 82808280
Sign −1-1
Analytic cond. 66.116166.1161
Root an. cond. 8.131188.13118
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  + 5-s − 2·7-s + 13-s + 4·17-s − 4·19-s − 23-s + 25-s + 3·29-s − 31-s − 2·35-s − 8·37-s + 5·41-s − 6·43-s − 9·47-s − 3·49-s − 2·53-s + 65-s + 4·67-s − 3·71-s + 7·73-s + 4·79-s − 8·83-s + 4·85-s + 14·89-s − 2·91-s − 4·95-s − 14·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.277·13-s + 0.970·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.557·29-s − 0.179·31-s − 0.338·35-s − 1.31·37-s + 0.780·41-s − 0.914·43-s − 1.31·47-s − 3/7·49-s − 0.274·53-s + 0.124·65-s + 0.488·67-s − 0.356·71-s + 0.819·73-s + 0.450·79-s − 0.878·83-s + 0.433·85-s + 1.48·89-s − 0.209·91-s − 0.410·95-s − 1.42·97-s + ⋯

Functional equation

Λ(s)=(8280s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8280s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 82808280    =    23⋅32⋅5⋅232^{3} \cdot 3^{2} \cdot 5 \cdot 23
Sign: −1-1
Analytic conductor: 66.116166.1161
Root analytic conductor: 8.131188.13118
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8280, ( :1/2), −1)(2,\ 8280,\ (\ :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
5 1−T 1 - T
23 1+T 1 + T
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−7.41828221426730488565657012205, −6.58312615606714474227509842902, −6.24103414973355131926441249927, −5.41411450162870433403656312461, −4.73502907055562708350414375027, −3.72156006813250260796216967867, −3.17490651892526508204222014921, −2.22114910796226392630475232768, −1.29434324689996725627123353605, 0, 1.29434324689996725627123353605, 2.22114910796226392630475232768, 3.17490651892526508204222014921, 3.72156006813250260796216967867, 4.73502907055562708350414375027, 5.41411450162870433403656312461, 6.24103414973355131926441249927, 6.58312615606714474227509842902, 7.41828221426730488565657012205

Graph of the ZZ-function along the critical line