↑

Properties

Label 2-416-1.1-c1-0-10
Degree 22
Conductor 416416
Sign −1-1
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 3·7-s − 2·9-s − 2·11-s + 13-s − 15-s − 3·17-s − 2·19-s + 3·21-s − 4·23-s − 4·25-s + 5·27-s + 2·29-s − 4·31-s + 2·33-s − 3·35-s + 5·37-s − 39-s − 12·41-s − 7·43-s − 2·45-s + 9·47-s + 2·49-s + 3·51-s + 4·53-s − 2·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.13·7-s − 2/3·9-s − 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.727·17-s − 0.458·19-s + 0.654·21-s − 0.834·23-s − 4/5·25-s + 0.962·27-s + 0.371·29-s − 0.718·31-s + 0.348·33-s − 0.507·35-s + 0.821·37-s − 0.160·39-s − 1.87·41-s − 1.06·43-s − 0.298·45-s + 1.31·47-s + 2/7·49-s + 0.420·51-s + 0.549·53-s − 0.269·55-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −1-1
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 416, ( :1/2), −1)(2,\ 416,\ (\ :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
13 1−T 1 - T
good3 1+T+pT2 1 + T + p T^{2}
5 1−T+pT2 1 - T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−5T+pT2 1 - 5 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1+7T+pT2 1 + 7 T + p T^{2}
47 1−9T+pT2 1 - 9 T + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1−15T+pT2 1 - 15 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p T^{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

−10.69730552679748697536416998477, −9.976912614651428361493366785064, −9.039528016718973273192662402272, −8.052297265441174562544922743090, −6.65408062467448396251062225874, −6.08119338191936199705626075673, −5.12564549383277447468893363985, −3.64849442843249490400769039530, −2.34351544305748892558428024728, 0, 2.34351544305748892558428024728, 3.64849442843249490400769039530, 5.12564549383277447468893363985, 6.08119338191936199705626075673, 6.65408062467448396251062225874, 8.052297265441174562544922743090, 9.039528016718973273192662402272, 9.976912614651428361493366785064, 10.69730552679748697536416998477

Graph of the ZZ-function along the critical line