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Properties

Label 2-416-1.1-c1-0-3
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  − 1.81·3-s + 2.70·5-s + 3.36·7-s + 0.298·9-s − 5.17·11-s + 13-s − 4.90·15-s + 6.70·17-s + 5.17·19-s − 6.10·21-s + 2.29·25-s + 4.90·27-s − 2·29-s + 8.80·31-s + 9.40·33-s + 9.08·35-s + 2.70·37-s − 1.81·39-s + 3.40·41-s − 8.53·43-s + 0.806·45-s + 3.36·47-s + 4.29·49-s − 12.1·51-s − 11.4·53-s − 13.9·55-s − 9.40·57-s + ⋯
L(s)  = 1  − 1.04·3-s + 1.20·5-s + 1.27·7-s + 0.0994·9-s − 1.56·11-s + 0.277·13-s − 1.26·15-s + 1.62·17-s + 1.18·19-s − 1.33·21-s + 0.459·25-s + 0.944·27-s − 0.371·29-s + 1.58·31-s + 1.63·33-s + 1.53·35-s + 0.444·37-s − 0.290·39-s + 0.531·41-s − 1.30·43-s + 0.120·45-s + 0.490·47-s + 0.614·49-s − 1.70·51-s − 1.56·53-s − 1.88·55-s − 1.24·57-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 1.3058075721.305807572
L(12)L(\frac12) ≈\approx 1.3058075721.305807572
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+1.81T+3T2 1 + 1.81T + 3T^{2}
5 1−2.70T+5T2 1 - 2.70T + 5T^{2}
7 1−3.36T+7T2 1 - 3.36T + 7T^{2}
11 1+5.17T+11T2 1 + 5.17T + 11T^{2}
17 1−6.70T+17T2 1 - 6.70T + 17T^{2}
19 1−5.17T+19T2 1 - 5.17T + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1−8.80T+31T2 1 - 8.80T + 31T^{2}
37 1−2.70T+37T2 1 - 2.70T + 37T^{2}
41 1−3.40T+41T2 1 - 3.40T + 41T^{2}
43 1+8.53T+43T2 1 + 8.53T + 43T^{2}
47 1−3.36T+47T2 1 - 3.36T + 47T^{2}
53 1+11.4T+53T2 1 + 11.4T + 53T^{2}
59 1−2.08T+59T2 1 - 2.08T + 59T^{2}
61 1+3.40T+61T2 1 + 3.40T + 61T^{2}
67 1−12.4T+67T2 1 - 12.4T + 67T^{2}
71 1+10.6T+71T2 1 + 10.6T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1+3.09T+79T2 1 + 3.09T + 79T^{2}
83 1−1.54T+83T2 1 - 1.54T + 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1+16.8T+97T2 1 + 16.8T + 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

−11.21769176412032772599458289663, −10.33269862446923915540639375074, −9.772728091282210140126435538215, −8.299780308373419818922978365132, −7.57968360952389312846678641449, −6.10301624159430199353302806935, −5.40340956474172915579499930517, −4.93494358162762048639693465362, −2.82690227683658217129719971052, −1.29802298639225669167687835755, 1.29802298639225669167687835755, 2.82690227683658217129719971052, 4.93494358162762048639693465362, 5.40340956474172915579499930517, 6.10301624159430199353302806935, 7.57968360952389312846678641449, 8.299780308373419818922978365132, 9.772728091282210140126435538215, 10.33269862446923915540639375074, 11.21769176412032772599458289663

Graph of the ZZ-function along the critical line