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Properties

Label 2-8280-1.1-c1-0-23
Degree 22
Conductor 82808280
Sign 11
Analytic cond. 66.116166.1161
Root an. cond. 8.131188.13118
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  − 5-s + 0.121·7-s − 2.87·11-s + 5.22·13-s + 2.22·17-s + 1.22·19-s − 23-s + 25-s − 9.34·29-s − 2.12·31-s − 0.121·35-s + 5.59·37-s − 8.22·41-s + 8·43-s + 10.4·47-s − 6.98·49-s + 3.59·53-s + 2.87·55-s + 0.650·59-s − 7.33·61-s − 5.22·65-s + 5.59·67-s + 13.9·71-s + 12.9·73-s − 0.349·77-s − 3.51·79-s + 11.1·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.0459·7-s − 0.867·11-s + 1.45·13-s + 0.540·17-s + 0.281·19-s − 0.208·23-s + 0.200·25-s − 1.73·29-s − 0.381·31-s − 0.0205·35-s + 0.919·37-s − 1.28·41-s + 1.21·43-s + 1.52·47-s − 0.997·49-s + 0.493·53-s + 0.388·55-s + 0.0846·59-s − 0.939·61-s − 0.648·65-s + 0.683·67-s + 1.65·71-s + 1.51·73-s − 0.0398·77-s − 0.395·79-s + 1.21·83-s + ⋯

Functional equation

Λ(s)=(8280s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8280s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 82808280    =    23⋅32⋅5⋅232^{3} \cdot 3^{2} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 66.116166.1161
Root analytic conductor: 8.131188.13118
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8280, ( :1/2), 1)(2,\ 8280,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7302498881.730249888
L(12)L(\frac12) ≈\approx 1.7302498881.730249888
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 1
5 1+T 1 + T
23 1+T 1 + T
good7 1−0.121T+7T2 1 - 0.121T + 7T^{2}
11 1+2.87T+11T2 1 + 2.87T + 11T^{2}
13 1−5.22T+13T2 1 - 5.22T + 13T^{2}
17 1−2.22T+17T2 1 - 2.22T + 17T^{2}
19 1−1.22T+19T2 1 - 1.22T + 19T^{2}
29 1+9.34T+29T2 1 + 9.34T + 29T^{2}
31 1+2.12T+31T2 1 + 2.12T + 31T^{2}
37 1−5.59T+37T2 1 - 5.59T + 37T^{2}
41 1+8.22T+41T2 1 + 8.22T + 41T^{2}
43 1−8T+43T2 1 - 8T + 43T^{2}
47 1−10.4T+47T2 1 - 10.4T + 47T^{2}
53 1−3.59T+53T2 1 - 3.59T + 53T^{2}
59 1−0.650T+59T2 1 - 0.650T + 59T^{2}
61 1+7.33T+61T2 1 + 7.33T + 61T^{2}
67 1−5.59T+67T2 1 - 5.59T + 67T^{2}
71 1−13.9T+71T2 1 - 13.9T + 71T^{2}
73 1−12.9T+73T2 1 - 12.9T + 73T^{2}
79 1+3.51T+79T2 1 + 3.51T + 79T^{2}
83 1−11.1T+83T2 1 - 11.1T + 83T^{2}
89 1−0.486T+89T2 1 - 0.486T + 89T^{2}
97 1−0.635T+97T2 1 - 0.635T + 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

−7.933941265653865905993035050915, −7.22639220921320099555227471987, −6.41010998119142004645101667310, −5.63013345753158074281748672730, −5.19332366885526801019576690191, −4.04440243182723350194702502142, −3.64142121406332749149034107718, −2.73133099074177703330270970874, −1.71872603270270747772551083490, −0.65559402291503074044996035287, 0.65559402291503074044996035287, 1.71872603270270747772551083490, 2.73133099074177703330270970874, 3.64142121406332749149034107718, 4.04440243182723350194702502142, 5.19332366885526801019576690191, 5.63013345753158074281748672730, 6.41010998119142004645101667310, 7.22639220921320099555227471987, 7.933941265653865905993035050915

Graph of the ZZ-function along the critical line