↑

Properties

Label 2-416-1.1-c1-0-7
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·3-s + 0.561·5-s − 0.561·7-s + 3.56·9-s + 2·11-s − 13-s + 1.43·15-s − 0.561·17-s + 6·19-s − 1.43·21-s − 4.68·25-s + 1.43·27-s − 8.24·29-s + 7.12·31-s + 5.12·33-s − 0.315·35-s − 9.68·37-s − 2.56·39-s + 7.12·41-s − 8.80·43-s + 2.00·45-s + 1.68·47-s − 6.68·49-s − 1.43·51-s − 4.87·53-s + 1.12·55-s + 15.3·57-s + ⋯
L(s)  = 1  + 1.47·3-s + 0.251·5-s − 0.212·7-s + 1.18·9-s + 0.603·11-s − 0.277·13-s + 0.371·15-s − 0.136·17-s + 1.37·19-s − 0.313·21-s − 0.936·25-s + 0.276·27-s − 1.53·29-s + 1.27·31-s + 0.891·33-s − 0.0533·35-s − 1.59·37-s − 0.410·39-s + 1.11·41-s − 1.34·43-s + 0.298·45-s + 0.245·47-s − 0.954·49-s − 0.201·51-s − 0.669·53-s + 0.151·55-s + 2.03·57-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 2.2389276472.238927647
L(12)L(\frac12) ≈\approx 2.2389276472.238927647
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 1−2.56T+3T2 1 - 2.56T + 3T^{2}
5 1−0.561T+5T2 1 - 0.561T + 5T^{2}
7 1+0.561T+7T2 1 + 0.561T + 7T^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
17 1+0.561T+17T2 1 + 0.561T + 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+8.24T+29T2 1 + 8.24T + 29T^{2}
31 1−7.12T+31T2 1 - 7.12T + 31T^{2}
37 1+9.68T+37T2 1 + 9.68T + 37T^{2}
41 1−7.12T+41T2 1 - 7.12T + 41T^{2}
43 1+8.80T+43T2 1 + 8.80T + 43T^{2}
47 1−1.68T+47T2 1 - 1.68T + 47T^{2}
53 1+4.87T+53T2 1 + 4.87T + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1−13.3T+61T2 1 - 13.3T + 61T^{2}
67 1−6T+67T2 1 - 6T + 67T^{2}
71 1+1.68T+71T2 1 + 1.68T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1−12T+79T2 1 - 12T + 79T^{2}
83 1+17.3T+83T2 1 + 17.3T + 83T^{2}
89 1+8.24T+89T2 1 + 8.24T + 89T^{2}
97 1+6T+97T2 1 + 6T + 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

−11.23461220324178498488048635367, −9.751879759476835518902712204197, −9.554414948318633638956058359001, −8.491922494420523496462340846268, −7.68638232220040779727430308657, −6.74646716726994014464223907671, −5.39667206829653893083510538697, −3.94563140609196492072449267321, −3.05843773369488483717877797781, −1.79757123529249810532635130140, 1.79757123529249810532635130140, 3.05843773369488483717877797781, 3.94563140609196492072449267321, 5.39667206829653893083510538697, 6.74646716726994014464223907671, 7.68638232220040779727430308657, 8.491922494420523496462340846268, 9.554414948318633638956058359001, 9.751879759476835518902712204197, 11.23461220324178498488048635367

Graph of the ZZ-function along the critical line