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Properties

Label 2-8512-1.1-c1-0-111
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  + 1.61·3-s + 4.23·5-s + 7-s − 0.381·9-s − 2.61·11-s − 3.47·13-s + 6.85·15-s − 5.85·17-s + 19-s + 1.61·21-s + 8.23·23-s + 12.9·25-s − 5.47·27-s + 4.61·29-s + 3.85·31-s − 4.23·33-s + 4.23·35-s + 8.23·37-s − 5.61·39-s + 11.5·41-s + 4.47·43-s − 1.61·45-s − 7.47·47-s + 49-s − 9.47·51-s − 6.09·53-s − 11.0·55-s + ⋯
L(s)  = 1  + 0.934·3-s + 1.89·5-s + 0.377·7-s − 0.127·9-s − 0.789·11-s − 0.962·13-s + 1.76·15-s − 1.41·17-s + 0.229·19-s + 0.353·21-s + 1.71·23-s + 2.58·25-s − 1.05·27-s + 0.857·29-s + 0.692·31-s − 0.737·33-s + 0.716·35-s + 1.35·37-s − 0.899·39-s + 1.80·41-s + 0.681·43-s − 0.241·45-s − 1.08·47-s + 0.142·49-s − 1.32·51-s − 0.836·53-s − 1.49·55-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 4.1896252794.189625279
L(12)L(\frac12) ≈\approx 4.1896252794.189625279
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 1−1.61T+3T2 1 - 1.61T + 3T^{2}
5 1−4.23T+5T2 1 - 4.23T + 5T^{2}
11 1+2.61T+11T2 1 + 2.61T + 11T^{2}
13 1+3.47T+13T2 1 + 3.47T + 13T^{2}
17 1+5.85T+17T2 1 + 5.85T + 17T^{2}
23 1−8.23T+23T2 1 - 8.23T + 23T^{2}
29 1−4.61T+29T2 1 - 4.61T + 29T^{2}
31 1−3.85T+31T2 1 - 3.85T + 31T^{2}
37 1−8.23T+37T2 1 - 8.23T + 37T^{2}
41 1−11.5T+41T2 1 - 11.5T + 41T^{2}
43 1−4.47T+43T2 1 - 4.47T + 43T^{2}
47 1+7.47T+47T2 1 + 7.47T + 47T^{2}
53 1+6.09T+53T2 1 + 6.09T + 53T^{2}
59 1−11T+59T2 1 - 11T + 59T^{2}
61 1−4.23T+61T2 1 - 4.23T + 61T^{2}
67 1+7.56T+67T2 1 + 7.56T + 67T^{2}
71 1−3.76T+71T2 1 - 3.76T + 71T^{2}
73 1−3.61T+73T2 1 - 3.61T + 73T^{2}
79 1−4.47T+79T2 1 - 4.47T + 79T^{2}
83 1−10.5T+83T2 1 - 10.5T + 83T^{2}
89 1+15.7T+89T2 1 + 15.7T + 89T^{2}
97 1+3.47T+97T2 1 + 3.47T + 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

−7.889763638627080904797154722078, −7.07666420384606181151944598699, −6.41597735150515538066262257814, −5.68564352726399266896976539462, −4.98482593387121022959827339150, −4.49207863582566691064157493999, −2.99929835083547694025198292649, −2.52451419717471534028990093556, −2.16272039062793630980081983116, −0.960471341176697098285565153514, 0.960471341176697098285565153514, 2.16272039062793630980081983116, 2.52451419717471534028990093556, 2.99929835083547694025198292649, 4.49207863582566691064157493999, 4.98482593387121022959827339150, 5.68564352726399266896976539462, 6.41597735150515538066262257814, 7.07666420384606181151944598699, 7.889763638627080904797154722078

Graph of the ZZ-function along the critical line