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Properties

Label 2-8040-1.1-c1-0-52
Degree 22
Conductor 80408040
Sign 11
Analytic cond. 64.199764.1997
Root an. cond. 8.012478.01247
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-s + 5-s − 2.42·7-s + 9-s + 2.63·11-s + 3.28·13-s + 15-s + 8.05·17-s − 1.54·19-s − 2.42·21-s + 1.83·23-s + 25-s + 27-s − 6.09·29-s + 5.70·31-s + 2.63·33-s − 2.42·35-s − 5.18·37-s + 3.28·39-s + 0.845·41-s − 1.75·43-s + 45-s + 9.75·47-s − 1.11·49-s + 8.05·51-s + 1.12·53-s + 2.63·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.917·7-s + 0.333·9-s + 0.792·11-s + 0.911·13-s + 0.258·15-s + 1.95·17-s − 0.354·19-s − 0.529·21-s + 0.382·23-s + 0.200·25-s + 0.192·27-s − 1.13·29-s + 1.02·31-s + 0.457·33-s − 0.410·35-s − 0.851·37-s + 0.526·39-s + 0.131·41-s − 0.267·43-s + 0.149·45-s + 1.42·47-s − 0.158·49-s + 1.12·51-s + 0.154·53-s + 0.354·55-s + ⋯

Functional equation

Λ(s)=(8040s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8040s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 80408040    =    23⋅3⋅5⋅672^{3} \cdot 3 \cdot 5 \cdot 67
Sign: 11
Analytic conductor: 64.199764.1997
Root analytic conductor: 8.012478.01247
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8040, ( :1/2), 1)(2,\ 8040,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1056562093.105656209
L(12)L(\frac12) ≈\approx 3.1056562093.105656209
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
3 1−T 1 - T
5 1−T 1 - T
67 1+T 1 + T
good7 1+2.42T+7T2 1 + 2.42T + 7T^{2}
11 1−2.63T+11T2 1 - 2.63T + 11T^{2}
13 1−3.28T+13T2 1 - 3.28T + 13T^{2}
17 1−8.05T+17T2 1 - 8.05T + 17T^{2}
19 1+1.54T+19T2 1 + 1.54T + 19T^{2}
23 1−1.83T+23T2 1 - 1.83T + 23T^{2}
29 1+6.09T+29T2 1 + 6.09T + 29T^{2}
31 1−5.70T+31T2 1 - 5.70T + 31T^{2}
37 1+5.18T+37T2 1 + 5.18T + 37T^{2}
41 1−0.845T+41T2 1 - 0.845T + 41T^{2}
43 1+1.75T+43T2 1 + 1.75T + 43T^{2}
47 1−9.75T+47T2 1 - 9.75T + 47T^{2}
53 1−1.12T+53T2 1 - 1.12T + 53T^{2}
59 1−0.551T+59T2 1 - 0.551T + 59T^{2}
61 1+5.56T+61T2 1 + 5.56T + 61T^{2}
71 1−13.9T+71T2 1 - 13.9T + 71T^{2}
73 1+14.9T+73T2 1 + 14.9T + 73T^{2}
79 1−16.7T+79T2 1 - 16.7T + 79T^{2}
83 1−4.42T+83T2 1 - 4.42T + 83T^{2}
89 1+10.1T+89T2 1 + 10.1T + 89T^{2}
97 1−1.04T+97T2 1 - 1.04T + 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

−7.84578409928363676599700467678, −7.13549228978471244231328140357, −6.40093373675074628398865373198, −5.89737029943140366717175136348, −5.12748668556653681262021443162, −3.98051456506430132190873111193, −3.49590961569892297952179998514, −2.83613994195869599740054970425, −1.72261113257134168632192950094, −0.899649525343639138748590313804, 0.899649525343639138748590313804, 1.72261113257134168632192950094, 2.83613994195869599740054970425, 3.49590961569892297952179998514, 3.98051456506430132190873111193, 5.12748668556653681262021443162, 5.89737029943140366717175136348, 6.40093373675074628398865373198, 7.13549228978471244231328140357, 7.84578409928363676599700467678

Graph of the ZZ-function along the critical line