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Properties

Label 2-460e2-1.1-c1-0-34
Degree 22
Conductor 211600211600
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s + 9-s − 2·13-s − 6·17-s − 4·19-s + 4·21-s + 4·27-s + 6·29-s + 4·31-s + 2·37-s + 4·39-s + 6·41-s + 10·43-s − 6·47-s − 3·49-s + 12·51-s − 6·53-s + 8·57-s − 12·59-s − 2·61-s − 2·63-s − 2·67-s + 12·71-s − 2·73-s + 8·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s + 0.640·39-s + 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s + 1.68·51-s − 0.824·53-s + 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.251·63-s − 0.244·67-s + 1.42·71-s − 0.234·73-s + 0.900·79-s − 1.22·81-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: −1-1
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: 11
Selberg data: (2, 211600, ( :1/2), −1)(2,\ 211600,\ (\ :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
5 1 1
23 1 1
good3 1+2T+pT2 1 + 2 T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+4T+pT2 1 + 4 T + 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 1−6T+pT2 1 - 6 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−12T+pT2 1 - 12 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+6T+pT2 1 + 6 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−13.00228418619092, −12.75665341872606, −12.23237309737730, −11.97314986962569, −11.24166099068490, −10.90445323361229, −10.67518142999101, −10.00027913760533, −9.555907532779929, −9.086226047208854, −8.569591601682723, −7.990499620368098, −7.470235081329580, −6.638108540374705, −6.547725124612121, −6.156470769739191, −5.614695357288647, −4.796884634118764, −4.635562562247647, −4.112463655446853, −3.269526651404290, −2.646934085891658, −2.264090339457877, −1.319394610645741, −0.5450275462336395, 0, 0.5450275462336395, 1.319394610645741, 2.264090339457877, 2.646934085891658, 3.269526651404290, 4.112463655446853, 4.635562562247647, 4.796884634118764, 5.614695357288647, 6.156470769739191, 6.547725124612121, 6.638108540374705, 7.470235081329580, 7.990499620368098, 8.569591601682723, 9.086226047208854, 9.555907532779929, 10.00027913760533, 10.67518142999101, 10.90445323361229, 11.24166099068490, 11.97314986962569, 12.23237309737730, 12.75665341872606, 13.00228418619092

Graph of the ZZ-function along the critical line