↑

Properties

Label 2-9200-1.1-c1-0-123
Degree 22
Conductor 92009200
Sign −1-1
Analytic cond. 73.462373.4623
Root an. cond. 8.571018.57101
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.631·3-s − 3.16·7-s − 2.60·9-s + 1.90·11-s + 6.76·13-s − 0.737·17-s − 3.90·19-s + 1.99·21-s − 23-s + 3.53·27-s − 0.935·29-s − 1.02·31-s − 1.20·33-s − 5.94·37-s − 4.27·39-s + 11.0·41-s − 3.90·43-s + 5.47·47-s + 3.03·49-s + 0.465·51-s − 3.07·53-s + 2.46·57-s − 4.90·59-s + 11.0·61-s + 8.24·63-s − 3.20·67-s + 0.631·69-s + ⋯
L(s)  = 1  − 0.364·3-s − 1.19·7-s − 0.867·9-s + 0.574·11-s + 1.87·13-s − 0.178·17-s − 0.895·19-s + 0.436·21-s − 0.208·23-s + 0.680·27-s − 0.173·29-s − 0.184·31-s − 0.209·33-s − 0.976·37-s − 0.684·39-s + 1.71·41-s − 0.595·43-s + 0.798·47-s + 0.433·49-s + 0.0651·51-s − 0.422·53-s + 0.326·57-s − 0.638·59-s + 1.41·61-s + 1.03·63-s − 0.391·67-s + 0.0760·69-s + ⋯

Functional equation

Λ(s)=(9200s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9200s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92009200    =    24⋅52⋅232^{4} \cdot 5^{2} \cdot 23
Sign: −1-1
Analytic conductor: 73.462373.4623
Root analytic conductor: 8.571018.57101
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9200, ( :1/2), −1)(2,\ 9200,\ (\ :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+T 1 + T
good3 1+0.631T+3T2 1 + 0.631T + 3T^{2}
7 1+3.16T+7T2 1 + 3.16T + 7T^{2}
11 1−1.90T+11T2 1 - 1.90T + 11T^{2}
13 1−6.76T+13T2 1 - 6.76T + 13T^{2}
17 1+0.737T+17T2 1 + 0.737T + 17T^{2}
19 1+3.90T+19T2 1 + 3.90T + 19T^{2}
29 1+0.935T+29T2 1 + 0.935T + 29T^{2}
31 1+1.02T+31T2 1 + 1.02T + 31T^{2}
37 1+5.94T+37T2 1 + 5.94T + 37T^{2}
41 1−11.0T+41T2 1 - 11.0T + 41T^{2}
43 1+3.90T+43T2 1 + 3.90T + 43T^{2}
47 1−5.47T+47T2 1 - 5.47T + 47T^{2}
53 1+3.07T+53T2 1 + 3.07T + 53T^{2}
59 1+4.90T+59T2 1 + 4.90T + 59T^{2}
61 1−11.0T+61T2 1 - 11.0T + 61T^{2}
67 1+3.20T+67T2 1 + 3.20T + 67T^{2}
71 1+8.86T+71T2 1 + 8.86T + 71T^{2}
73 1−9.40T+73T2 1 - 9.40T + 73T^{2}
79 1+1.56T+79T2 1 + 1.56T + 79T^{2}
83 1+2.64T+83T2 1 + 2.64T + 83T^{2}
89 1+18.6T+89T2 1 + 18.6T + 89T^{2}
97 1−15.3T+97T2 1 - 15.3T + 97T^{2}
show more
show less
   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

−7.19129645928260866732251439269, −6.47269429103235403811260766061, −6.07723354940678862060598793007, −5.63566386880028614419109092184, −4.48602814062654668803295152334, −3.71163170625325123692764237984, −3.23475953009717058303983811039, −2.22349093226576460215063467765, −1.07060145956354884422617677565, 0, 1.07060145956354884422617677565, 2.22349093226576460215063467765, 3.23475953009717058303983811039, 3.71163170625325123692764237984, 4.48602814062654668803295152334, 5.63566386880028614419109092184, 6.07723354940678862060598793007, 6.47269429103235403811260766061, 7.19129645928260866732251439269

Graph of the ZZ-function along the critical line