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Properties

Label 2-460e2-1.1-c1-0-46
Degree 22
Conductor 211600211600
Sign 11
Analytic cond. 1689.631689.63
Root an. cond. 41.105141.1051
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  + 2·3-s + 7-s + 9-s − 5·11-s + 13-s + 4·17-s + 7·19-s + 2·21-s − 4·27-s + 5·29-s − 2·31-s − 10·33-s + 2·37-s + 2·39-s + 11·41-s + 43-s + 8·47-s − 6·49-s + 8·51-s + 14·57-s + 14·59-s − 10·61-s + 63-s − 8·67-s + 10·71-s + 7·73-s − 5·77-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.277·13-s + 0.970·17-s + 1.60·19-s + 0.436·21-s − 0.769·27-s + 0.928·29-s − 0.359·31-s − 1.74·33-s + 0.328·37-s + 0.320·39-s + 1.71·41-s + 0.152·43-s + 1.16·47-s − 6/7·49-s + 1.12·51-s + 1.85·57-s + 1.82·59-s − 1.28·61-s + 0.125·63-s − 0.977·67-s + 1.18·71-s + 0.819·73-s − 0.569·77-s + ⋯

Functional equation

Λ(s)=(211600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(211600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 211600211600    =    24⋅52⋅2322^{4} \cdot 5^{2} \cdot 23^{2}
Sign: 11
Analytic conductor: 1689.631689.63
Root analytic conductor: 41.105141.1051
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 211600, ( :1/2), 1)(2,\ 211600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.8665511654.866551165
L(12)L(\frac12) ≈\approx 4.8665511654.866551165
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
5 1 1
23 1 1
good3 1−2T+pT2 1 - 2 T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
29 1−5T+pT2 1 - 5 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−11T+pT2 1 - 11 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−14T+pT2 1 - 14 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1−7T+pT2 1 - 7 T + p T^{2}
83 1+15T+pT2 1 + 15 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1−4T+pT2 1 - 4 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

−13.00745842626360, −12.72330528525986, −12.10753974962872, −11.64044209198918, −11.04896208724363, −10.70255522775490, −10.03818389211253, −9.691006886434241, −9.295318758118139, −8.583238680411013, −8.319830526436510, −7.728158385698417, −7.522891486471332, −7.124097115579591, −6.086339680737491, −5.744325183831230, −5.210884622447702, −4.753778601371603, −4.021405350605338, −3.439717847452019, −2.956359360569365, −2.590438113327234, −2.003456823794605, −1.171779490024840, −0.6070515297217679, 0.6070515297217679, 1.171779490024840, 2.003456823794605, 2.590438113327234, 2.956359360569365, 3.439717847452019, 4.021405350605338, 4.753778601371603, 5.210884622447702, 5.744325183831230, 6.086339680737491, 7.124097115579591, 7.522891486471332, 7.728158385698417, 8.319830526436510, 8.583238680411013, 9.295318758118139, 9.691006886434241, 10.03818389211253, 10.70255522775490, 11.04896208724363, 11.64044209198918, 12.10753974962872, 12.72330528525986, 13.00745842626360

Graph of the ZZ-function along the critical line