↑

Properties

Label 2-920-1.1-c1-0-21
Degree 22
Conductor 920920
Sign −1-1
Analytic cond. 7.346237.34623
Root an. cond. 2.710392.71039
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  + 1.56·3-s − 5-s − 2.56·7-s − 0.561·9-s − 2·11-s − 3.56·13-s − 1.56·15-s + 1.43·17-s − 2·19-s − 4·21-s − 23-s + 25-s − 5.56·27-s − 8.12·29-s + 0.123·31-s − 3.12·33-s + 2.56·35-s + 0.561·37-s − 5.56·39-s + 4.12·41-s + 6.24·43-s + 0.561·45-s − 8.68·47-s − 0.438·49-s + 2.24·51-s + 8.56·53-s + 2·55-s + ⋯
L(s)  = 1  + 0.901·3-s − 0.447·5-s − 0.968·7-s − 0.187·9-s − 0.603·11-s − 0.987·13-s − 0.403·15-s + 0.348·17-s − 0.458·19-s − 0.872·21-s − 0.208·23-s + 0.200·25-s − 1.07·27-s − 1.50·29-s + 0.0221·31-s − 0.543·33-s + 0.432·35-s + 0.0923·37-s − 0.890·39-s + 0.643·41-s + 0.952·43-s + 0.0837·45-s − 1.26·47-s − 0.0626·49-s + 0.314·51-s + 1.17·53-s + 0.269·55-s + ⋯

Functional equation

Λ(s)=(920s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(920s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 920920    =    23⋅5⋅232^{3} \cdot 5 \cdot 23
Sign: −1-1
Analytic conductor: 7.346237.34623
Root analytic conductor: 2.710392.71039
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 920, ( :1/2), −1)(2,\ 920,\ (\ :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+T 1 + T
23 1+T 1 + T
good3 1−1.56T+3T2 1 - 1.56T + 3T^{2}
7 1+2.56T+7T2 1 + 2.56T + 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1+3.56T+13T2 1 + 3.56T + 13T^{2}
17 1−1.43T+17T2 1 - 1.43T + 17T^{2}
19 1+2T+19T2 1 + 2T + 19T^{2}
29 1+8.12T+29T2 1 + 8.12T + 29T^{2}
31 1−0.123T+31T2 1 - 0.123T + 31T^{2}
37 1−0.561T+37T2 1 - 0.561T + 37T^{2}
41 1−4.12T+41T2 1 - 4.12T + 41T^{2}
43 1−6.24T+43T2 1 - 6.24T + 43T^{2}
47 1+8.68T+47T2 1 + 8.68T + 47T^{2}
53 1−8.56T+53T2 1 - 8.56T + 53T^{2}
59 1+1.43T+59T2 1 + 1.43T + 59T^{2}
61 1+0.876T+61T2 1 + 0.876T + 61T^{2}
67 1+7.43T+67T2 1 + 7.43T + 67T^{2}
71 1+5T+71T2 1 + 5T + 71T^{2}
73 1−7.56T+73T2 1 - 7.56T + 73T^{2}
79 1+0.876T+79T2 1 + 0.876T + 79T^{2}
83 1−7.68T+83T2 1 - 7.68T + 83T^{2}
89 1−8T+89T2 1 - 8T + 89T^{2}
97 1−5.12T+97T2 1 - 5.12T + 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

−9.527700492840389776085760445404, −8.918055005191489921962297743259, −7.87787094055281652610846232806, −7.41800561273076131734096444241, −6.26167348608412767415468072335, −5.25635349558868339763553177976, −3.99454410774820647075962821105, −3.11920822435374775943149076422, −2.27445453388986391326208563508, 0, 2.27445453388986391326208563508, 3.11920822435374775943149076422, 3.99454410774820647075962821105, 5.25635349558868339763553177976, 6.26167348608412767415468072335, 7.41800561273076131734096444241, 7.87787094055281652610846232806, 8.918055005191489921962297743259, 9.527700492840389776085760445404

Graph of the ZZ-function along the critical line