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Properties

Label 2-460e2-1.1-c1-0-21
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  − 7-s − 3·9-s − 5·11-s − 5·13-s − 6·17-s − 5·19-s + 3·29-s + 4·31-s − 6·37-s − 9·41-s + 5·43-s − 10·47-s − 6·49-s − 12·53-s − 2·59-s + 10·61-s + 3·63-s + 4·67-s − 6·71-s + 15·73-s + 5·77-s − 15·79-s + 9·81-s − 7·83-s + 10·89-s + 5·91-s − 10·97-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s − 1.50·11-s − 1.38·13-s − 1.45·17-s − 1.14·19-s + 0.557·29-s + 0.718·31-s − 0.986·37-s − 1.40·41-s + 0.762·43-s − 1.45·47-s − 6/7·49-s − 1.64·53-s − 0.260·59-s + 1.28·61-s + 0.377·63-s + 0.488·67-s − 0.712·71-s + 1.75·73-s + 0.569·77-s − 1.68·79-s + 81-s − 0.768·83-s + 1.05·89-s + 0.524·91-s − 1.01·97-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+pT2 1 + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1−5T+pT2 1 - 5 T + p T^{2}
47 1+10T+pT2 1 + 10 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+2T+pT2 1 + 2 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1−15T+pT2 1 - 15 T + p T^{2}
79 1+15T+pT2 1 + 15 T + p T^{2}
83 1+7T+pT2 1 + 7 T + p T^{2}
89 1−10T+pT2 1 - 10 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

−13.20743165390675, −12.76666353670297, −12.42149132188081, −11.86944983051091, −11.23373765550092, −11.01229967010054, −10.40297359976151, −9.962216225209009, −9.641869440099252, −8.834470196512242, −8.534524784953805, −8.084879611488656, −7.648459765520313, −6.851050349243827, −6.614952490237457, −6.098186485662438, −5.365183268674080, −4.874630468628633, −4.724358466661874, −3.876790646030350, −3.093571558391978, −2.732435491183429, −2.265736090756400, −1.723966170734502, −0.3680168119170131, 0, 0.3680168119170131, 1.723966170734502, 2.265736090756400, 2.732435491183429, 3.093571558391978, 3.876790646030350, 4.724358466661874, 4.874630468628633, 5.365183268674080, 6.098186485662438, 6.614952490237457, 6.851050349243827, 7.648459765520313, 8.084879611488656, 8.534524784953805, 8.834470196512242, 9.641869440099252, 9.962216225209009, 10.40297359976151, 11.01229967010054, 11.23373765550092, 11.86944983051091, 12.42149132188081, 12.76666353670297, 13.20743165390675

Graph of the ZZ-function along the critical line