↑

Properties

Label 2-968-1.1-c1-0-21
Degree 22
Conductor 968968
Sign −1-1
Analytic cond. 7.729517.72951
Root an. cond. 2.780202.78020
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  + 0.618·3-s − 1.61·5-s − 0.618·7-s − 2.61·9-s + 0.854·13-s − 1.00·15-s + 4.85·17-s − 1.85·19-s − 0.381·21-s − 4·23-s − 2.38·25-s − 3.47·27-s − 8.32·29-s − 10.0·31-s + 1.00·35-s − 7.38·37-s + 0.527·39-s − 7.38·41-s + 10.4·43-s + 4.23·45-s + 9.56·47-s − 6.61·49-s + 3.00·51-s − 1.61·53-s − 1.14·57-s − 4.61·59-s + 4.85·61-s + ⋯
L(s)  = 1  + 0.356·3-s − 0.723·5-s − 0.233·7-s − 0.872·9-s + 0.236·13-s − 0.258·15-s + 1.17·17-s − 0.425·19-s − 0.0833·21-s − 0.834·23-s − 0.476·25-s − 0.668·27-s − 1.54·29-s − 1.81·31-s + 0.169·35-s − 1.21·37-s + 0.0845·39-s − 1.15·41-s + 1.59·43-s + 0.631·45-s + 1.39·47-s − 0.945·49-s + 0.420·51-s − 0.222·53-s − 0.151·57-s − 0.601·59-s + 0.621·61-s + ⋯

Functional equation

Λ(s)=(968s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(968s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 968968    =    23⋅1122^{3} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 7.729517.72951
Root analytic conductor: 2.780202.78020
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 968, ( :1/2), −1)(2,\ 968,\ (\ :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
11 1 1
good3 1−0.618T+3T2 1 - 0.618T + 3T^{2}
5 1+1.61T+5T2 1 + 1.61T + 5T^{2}
7 1+0.618T+7T2 1 + 0.618T + 7T^{2}
13 1−0.854T+13T2 1 - 0.854T + 13T^{2}
17 1−4.85T+17T2 1 - 4.85T + 17T^{2}
19 1+1.85T+19T2 1 + 1.85T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1+8.32T+29T2 1 + 8.32T + 29T^{2}
31 1+10.0T+31T2 1 + 10.0T + 31T^{2}
37 1+7.38T+37T2 1 + 7.38T + 37T^{2}
41 1+7.38T+41T2 1 + 7.38T + 41T^{2}
43 1−10.4T+43T2 1 - 10.4T + 43T^{2}
47 1−9.56T+47T2 1 - 9.56T + 47T^{2}
53 1+1.61T+53T2 1 + 1.61T + 53T^{2}
59 1+4.61T+59T2 1 + 4.61T + 59T^{2}
61 1−4.85T+61T2 1 - 4.85T + 61T^{2}
67 1+5.52T+67T2 1 + 5.52T + 67T^{2}
71 1−6.09T+71T2 1 - 6.09T + 71T^{2}
73 1+9.85T+73T2 1 + 9.85T + 73T^{2}
79 1−2.85T+79T2 1 - 2.85T + 79T^{2}
83 1−11.6T+83T2 1 - 11.6T + 83T^{2}
89 1−4.47T+89T2 1 - 4.47T + 89T^{2}
97 1−3.32T+97T2 1 - 3.32T + 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

−9.440641779066685401342939046575, −8.771811553898020880839308821769, −7.86307467986172443645176153073, −7.35278018492927500390542851351, −6.02829909448762209990088214931, −5.36704958470025533475891292204, −3.88477704108596484249207887343, −3.38218130462768167333156556197, −1.98237740972733751436662819945, 0, 1.98237740972733751436662819945, 3.38218130462768167333156556197, 3.88477704108596484249207887343, 5.36704958470025533475891292204, 6.02829909448762209990088214931, 7.35278018492927500390542851351, 7.86307467986172443645176153073, 8.771811553898020880839308821769, 9.440641779066685401342939046575

Graph of the ZZ-function along the critical line