↑

Properties

Label 2-4840-1.1-c1-0-33
Degree 22
Conductor 48404840
Sign 11
Analytic cond. 38.647538.6475
Root an. cond. 6.216716.21671
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·3-s − 5-s − 4.56·7-s + 3.56·9-s + 1.12·13-s − 2.56·15-s + 7.68·17-s − 1.43·19-s − 11.6·21-s + 1.12·23-s + 25-s + 1.43·27-s − 8.56·29-s − 1.43·31-s + 4.56·35-s + 7.43·37-s + 2.87·39-s + 12.2·41-s + 3.12·43-s − 3.56·45-s − 11.3·47-s + 13.8·49-s + 19.6·51-s + 9.68·53-s − 3.68·57-s + 1.12·59-s + 12.5·61-s + ⋯
L(s)  = 1  + 1.47·3-s − 0.447·5-s − 1.72·7-s + 1.18·9-s + 0.311·13-s − 0.661·15-s + 1.86·17-s − 0.330·19-s − 2.54·21-s + 0.234·23-s + 0.200·25-s + 0.276·27-s − 1.58·29-s − 0.258·31-s + 0.771·35-s + 1.22·37-s + 0.460·39-s + 1.91·41-s + 0.476·43-s − 0.530·45-s − 1.65·47-s + 1.97·49-s + 2.75·51-s + 1.33·53-s − 0.488·57-s + 0.146·59-s + 1.60·61-s + ⋯

Functional equation

Λ(s)=(4840s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4840s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 48404840    =    23⋅5⋅1122^{3} \cdot 5 \cdot 11^{2}
Sign: 11
Analytic conductor: 38.647538.6475
Root analytic conductor: 6.216716.21671
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4840, ( :1/2), 1)(2,\ 4840,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5353126532.535312653
L(12)L(\frac12) ≈\approx 2.5353126532.535312653
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+T 1 + T
11 1 1
good3 1−2.56T+3T2 1 - 2.56T + 3T^{2}
7 1+4.56T+7T2 1 + 4.56T + 7T^{2}
13 1−1.12T+13T2 1 - 1.12T + 13T^{2}
17 1−7.68T+17T2 1 - 7.68T + 17T^{2}
19 1+1.43T+19T2 1 + 1.43T + 19T^{2}
23 1−1.12T+23T2 1 - 1.12T + 23T^{2}
29 1+8.56T+29T2 1 + 8.56T + 29T^{2}
31 1+1.43T+31T2 1 + 1.43T + 31T^{2}
37 1−7.43T+37T2 1 - 7.43T + 37T^{2}
41 1−12.2T+41T2 1 - 12.2T + 41T^{2}
43 1−3.12T+43T2 1 - 3.12T + 43T^{2}
47 1+11.3T+47T2 1 + 11.3T + 47T^{2}
53 1−9.68T+53T2 1 - 9.68T + 53T^{2}
59 1−1.12T+59T2 1 - 1.12T + 59T^{2}
61 1−12.5T+61T2 1 - 12.5T + 61T^{2}
67 1+67T2 1 + 67T^{2}
71 1−3.68T+71T2 1 - 3.68T + 71T^{2}
73 1+1.12T+73T2 1 + 1.12T + 73T^{2}
79 1−11.3T+79T2 1 - 11.3T + 79T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1−9.68T+89T2 1 - 9.68T + 89T^{2}
97 1+4.87T+97T2 1 + 4.87T + 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

−8.192867505434845688401446029270, −7.67278783161813013540747539820, −7.07351747991955619339429976468, −6.16149101367801538889825010866, −5.47449664398454380934839962458, −4.04834404435399700897344942493, −3.59446625467074219122793148662, −3.04590249400907629988524953309, −2.23907318683614832488887099808, −0.804036707494051472523679283186, 0.804036707494051472523679283186, 2.23907318683614832488887099808, 3.04590249400907629988524953309, 3.59446625467074219122793148662, 4.04834404435399700897344942493, 5.47449664398454380934839962458, 6.16149101367801538889825010866, 7.07351747991955619339429976468, 7.67278783161813013540747539820, 8.192867505434845688401446029270

Graph of the ZZ-function along the critical line