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Properties

Label 2-8512-1.1-c1-0-13
Degree 22
Conductor 85128512
Sign 11
Analytic cond. 67.968667.9686
Root an. cond. 8.244318.24431
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  − 0.381·3-s + 5-s − 7-s − 2.85·9-s − 5.09·11-s − 2.23·13-s − 0.381·15-s − 5.38·17-s − 19-s + 0.381·21-s − 4.70·23-s − 4·25-s + 2.23·27-s + 5.85·29-s + 4.61·31-s + 1.94·33-s − 35-s − 6.70·37-s + 0.854·39-s − 11.0·41-s − 0.472·43-s − 2.85·45-s + 8.70·47-s + 49-s + 2.05·51-s − 1.32·53-s − 5.09·55-s + ⋯
L(s)  = 1  − 0.220·3-s + 0.447·5-s − 0.377·7-s − 0.951·9-s − 1.53·11-s − 0.620·13-s − 0.0986·15-s − 1.30·17-s − 0.229·19-s + 0.0833·21-s − 0.981·23-s − 0.800·25-s + 0.430·27-s + 1.08·29-s + 0.829·31-s + 0.338·33-s − 0.169·35-s − 1.10·37-s + 0.136·39-s − 1.73·41-s − 0.0720·43-s − 0.425·45-s + 1.27·47-s + 0.142·49-s + 0.287·51-s − 0.182·53-s − 0.686·55-s + ⋯

Functional equation

Λ(s)=(8512s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8512s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 85128512    =    26⋅7⋅192^{6} \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 67.968667.9686
Root analytic conductor: 8.244318.24431
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8512, ( :1/2), 1)(2,\ 8512,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.54114873080.5411487308
L(12)L(\frac12) ≈\approx 0.54114873080.5411487308
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
7 1+T 1 + T
19 1+T 1 + T
good3 1+0.381T+3T2 1 + 0.381T + 3T^{2}
5 1−T+5T2 1 - T + 5T^{2}
11 1+5.09T+11T2 1 + 5.09T + 11T^{2}
13 1+2.23T+13T2 1 + 2.23T + 13T^{2}
17 1+5.38T+17T2 1 + 5.38T + 17T^{2}
23 1+4.70T+23T2 1 + 4.70T + 23T^{2}
29 1−5.85T+29T2 1 - 5.85T + 29T^{2}
31 1−4.61T+31T2 1 - 4.61T + 31T^{2}
37 1+6.70T+37T2 1 + 6.70T + 37T^{2}
41 1+11.0T+41T2 1 + 11.0T + 41T^{2}
43 1+0.472T+43T2 1 + 0.472T + 43T^{2}
47 1−8.70T+47T2 1 - 8.70T + 47T^{2}
53 1+1.32T+53T2 1 + 1.32T + 53T^{2}
59 1−2.70T+59T2 1 - 2.70T + 59T^{2}
61 1−3.47T+61T2 1 - 3.47T + 61T^{2}
67 1−9.09T+67T2 1 - 9.09T + 67T^{2}
71 1+11.1T+71T2 1 + 11.1T + 71T^{2}
73 1+12.0T+73T2 1 + 12.0T + 73T^{2}
79 1+10.9T+79T2 1 + 10.9T + 79T^{2}
83 1−12.3T+83T2 1 - 12.3T + 83T^{2}
89 1−10.1T+89T2 1 - 10.1T + 89T^{2}
97 1+4.70T+97T2 1 + 4.70T + 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

−7.84603463354390528991459396507, −7.01325854932454351956224155297, −6.32797983846667977445671949667, −5.72798736689698697143935543489, −5.07822728077104275182769266687, −4.43814594818679021550988964113, −3.31900225799846868766661896904, −2.52836848408661493356716141149, −2.04548452184538125360581546211, −0.33264512993748612238798931157, 0.33264512993748612238798931157, 2.04548452184538125360581546211, 2.52836848408661493356716141149, 3.31900225799846868766661896904, 4.43814594818679021550988964113, 5.07822728077104275182769266687, 5.72798736689698697143935543489, 6.32797983846667977445671949667, 7.01325854932454351956224155297, 7.84603463354390528991459396507

Graph of the ZZ-function along the critical line