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Properties

Label 2-416-1.1-c1-0-5
Degree 22
Conductor 416416
Sign 11
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.11·3-s − 3.70·5-s + 4.20·7-s + 6.70·9-s − 1.09·11-s + 13-s − 11.5·15-s + 0.298·17-s + 1.09·19-s + 13.1·21-s + 8.70·25-s + 11.5·27-s − 2·29-s − 5.13·31-s − 3.40·33-s − 15.5·35-s − 3.70·37-s + 3.11·39-s − 9.40·41-s − 5.29·43-s − 24.8·45-s + 4.20·47-s + 10.7·49-s + 0.929·51-s + 1.40·53-s + 4.04·55-s + 3.40·57-s + ⋯
L(s)  = 1  + 1.79·3-s − 1.65·5-s + 1.59·7-s + 2.23·9-s − 0.329·11-s + 0.277·13-s − 2.97·15-s + 0.0723·17-s + 0.250·19-s + 2.85·21-s + 1.74·25-s + 2.21·27-s − 0.371·29-s − 0.922·31-s − 0.592·33-s − 2.63·35-s − 0.608·37-s + 0.498·39-s − 1.46·41-s − 0.808·43-s − 3.69·45-s + 0.613·47-s + 1.52·49-s + 0.130·51-s + 0.192·53-s + 0.545·55-s + 0.450·57-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: 11
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 416, ( :1/2), 1)(2,\ 416,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1925737872.192573787
L(12)L(\frac12) ≈\approx 2.1925737872.192573787
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 1−3.11T+3T2 1 - 3.11T + 3T^{2}
5 1+3.70T+5T2 1 + 3.70T + 5T^{2}
7 1−4.20T+7T2 1 - 4.20T + 7T^{2}
11 1+1.09T+11T2 1 + 1.09T + 11T^{2}
17 1−0.298T+17T2 1 - 0.298T + 17T^{2}
19 1−1.09T+19T2 1 - 1.09T + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+5.13T+31T2 1 + 5.13T + 31T^{2}
37 1+3.70T+37T2 1 + 3.70T + 37T^{2}
41 1+9.40T+41T2 1 + 9.40T + 41T^{2}
43 1+5.29T+43T2 1 + 5.29T + 43T^{2}
47 1−4.20T+47T2 1 - 4.20T + 47T^{2}
53 1−1.40T+53T2 1 - 1.40T + 53T^{2}
59 1+13.5T+59T2 1 + 13.5T + 59T^{2}
61 1−9.40T+61T2 1 - 9.40T + 61T^{2}
67 1+11.3T+67T2 1 + 11.3T + 67T^{2}
71 1−8.25T+71T2 1 - 8.25T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1+14.6T+79T2 1 + 14.6T + 79T^{2}
83 1−7.32T+83T2 1 - 7.32T + 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1−8.80T+97T2 1 - 8.80T + 97T^{2}
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   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

−11.27144267912799668933374417498, −10.31740728968048317657341225980, −8.894848668387295020147989288852, −8.408628391007486813206208698121, −7.70983541854264640308889868248, −7.23772232511848076950745793511, −4.95922426227104246141752968546, −4.04611840089777690187100592119, −3.20793014823390904891839771651, −1.72716761869207738952843222092, 1.72716761869207738952843222092, 3.20793014823390904891839771651, 4.04611840089777690187100592119, 4.95922426227104246141752968546, 7.23772232511848076950745793511, 7.70983541854264640308889868248, 8.408628391007486813206208698121, 8.894848668387295020147989288852, 10.31740728968048317657341225980, 11.27144267912799668933374417498

Graph of the ZZ-function along the critical line