↑

Properties

Label 2-968-1.1-c1-0-6
Degree 22
Conductor 968968
Sign 11
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 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.23·3-s + 2.23·5-s + 3.23·7-s + 7.47·9-s + 1.76·13-s − 7.23·15-s + 17-s − 5.70·19-s − 10.4·21-s + 0.763·23-s − 14.4·27-s + 1.76·29-s + 4.76·31-s + 7.23·35-s − 0.236·37-s − 5.70·39-s + 7.47·41-s + 10.4·43-s + 16.7·45-s + 5.70·47-s + 3.47·49-s − 3.23·51-s − 13.1·53-s + 18.4·57-s − 5.52·59-s + 14.9·61-s + 24.1·63-s + ⋯
L(s)  = 1  − 1.86·3-s + 0.999·5-s + 1.22·7-s + 2.49·9-s + 0.489·13-s − 1.86·15-s + 0.242·17-s − 1.30·19-s − 2.28·21-s + 0.159·23-s − 2.78·27-s + 0.327·29-s + 0.855·31-s + 1.22·35-s − 0.0388·37-s − 0.914·39-s + 1.16·41-s + 1.59·43-s + 2.49·45-s + 0.832·47-s + 0.496·49-s − 0.453·51-s − 1.81·53-s + 2.44·57-s − 0.719·59-s + 1.91·61-s + 3.04·63-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: 11
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: 00
Selberg data: (2, 968, ( :1/2), 1)(2,\ 968,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2246273431.224627343
L(12)L(\frac12) ≈\approx 1.2246273431.224627343
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+3.23T+3T2 1 + 3.23T + 3T^{2}
5 1−2.23T+5T2 1 - 2.23T + 5T^{2}
7 1−3.23T+7T2 1 - 3.23T + 7T^{2}
13 1−1.76T+13T2 1 - 1.76T + 13T^{2}
17 1−T+17T2 1 - T + 17T^{2}
19 1+5.70T+19T2 1 + 5.70T + 19T^{2}
23 1−0.763T+23T2 1 - 0.763T + 23T^{2}
29 1−1.76T+29T2 1 - 1.76T + 29T^{2}
31 1−4.76T+31T2 1 - 4.76T + 31T^{2}
37 1+0.236T+37T2 1 + 0.236T + 37T^{2}
41 1−7.47T+41T2 1 - 7.47T + 41T^{2}
43 1−10.4T+43T2 1 - 10.4T + 43T^{2}
47 1−5.70T+47T2 1 - 5.70T + 47T^{2}
53 1+13.1T+53T2 1 + 13.1T + 53T^{2}
59 1+5.52T+59T2 1 + 5.52T + 59T^{2}
61 1−14.9T+61T2 1 - 14.9T + 61T^{2}
67 1+0.763T+67T2 1 + 0.763T + 67T^{2}
71 1+4T+71T2 1 + 4T + 71T^{2}
73 1−3.52T+73T2 1 - 3.52T + 73T^{2}
79 1+7.23T+79T2 1 + 7.23T + 79T^{2}
83 1−12.1T+83T2 1 - 12.1T + 83T^{2}
89 1+12.4T+89T2 1 + 12.4T + 89T^{2}
97 1−11.9T+97T2 1 - 11.9T + 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

−10.31846294866429154463528586243, −9.446177358737667417530547865169, −8.288445990407859052952823952227, −7.26108951946828757838567593700, −6.19674225091174843181567445059, −5.86114259675181829736928730372, −4.89308847686214655907872192021, −4.25464656987388789999573726974, −2.07899325679545239886163437291, −1.02521012388011631360463105375, 1.02521012388011631360463105375, 2.07899325679545239886163437291, 4.25464656987388789999573726974, 4.89308847686214655907872192021, 5.86114259675181829736928730372, 6.19674225091174843181567445059, 7.26108951946828757838567593700, 8.288445990407859052952823952227, 9.446177358737667417530547865169, 10.31846294866429154463528586243

Graph of the ZZ-function along the critical line