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Properties

Label 2-8512-1.1-c1-0-11
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  − 1.79·3-s − 3·5-s + 7-s + 0.208·9-s − 0.791·11-s + 13-s + 5.37·15-s − 0.791·17-s − 19-s − 1.79·21-s − 4.58·23-s + 4·25-s + 5.00·27-s + 0.791·29-s − 6.37·31-s + 1.41·33-s − 3·35-s − 5·37-s − 1.79·39-s + 0.791·41-s − 2·43-s − 0.626·45-s + 1.41·47-s + 49-s + 1.41·51-s − 5.37·53-s + 2.37·55-s + ⋯
L(s)  = 1  − 1.03·3-s − 1.34·5-s + 0.377·7-s + 0.0695·9-s − 0.238·11-s + 0.277·13-s + 1.38·15-s − 0.191·17-s − 0.229·19-s − 0.390·21-s − 0.955·23-s + 0.800·25-s + 0.962·27-s + 0.146·29-s − 1.14·31-s + 0.246·33-s − 0.507·35-s − 0.821·37-s − 0.286·39-s + 0.123·41-s − 0.304·43-s − 0.0933·45-s + 0.206·47-s + 0.142·49-s + 0.198·51-s − 0.738·53-s + 0.320·55-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 0.35694124920.3569412492
L(12)L(\frac12) ≈\approx 0.35694124920.3569412492
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+1.79T+3T2 1 + 1.79T + 3T^{2}
5 1+3T+5T2 1 + 3T + 5T^{2}
11 1+0.791T+11T2 1 + 0.791T + 11T^{2}
13 1−T+13T2 1 - T + 13T^{2}
17 1+0.791T+17T2 1 + 0.791T + 17T^{2}
23 1+4.58T+23T2 1 + 4.58T + 23T^{2}
29 1−0.791T+29T2 1 - 0.791T + 29T^{2}
31 1+6.37T+31T2 1 + 6.37T + 31T^{2}
37 1+5T+37T2 1 + 5T + 37T^{2}
41 1−0.791T+41T2 1 - 0.791T + 41T^{2}
43 1+2T+43T2 1 + 2T + 43T^{2}
47 1−1.41T+47T2 1 - 1.41T + 47T^{2}
53 1+5.37T+53T2 1 + 5.37T + 53T^{2}
59 1−6.16T+59T2 1 - 6.16T + 59T^{2}
61 1−T+61T2 1 - T + 61T^{2}
67 1+4.37T+67T2 1 + 4.37T + 67T^{2}
71 1−6.16T+71T2 1 - 6.16T + 71T^{2}
73 1−2.62T+73T2 1 - 2.62T + 73T^{2}
79 1+10T+79T2 1 + 10T + 79T^{2}
83 1+0.626T+83T2 1 + 0.626T + 83T^{2}
89 1+1.58T+89T2 1 + 1.58T + 89T^{2}
97 1+7T+97T2 1 + 7T + 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

−7.81504786353188156896870995315, −7.05238747727061286609432930959, −6.44792033160151507544490550853, −5.59535575758865086949932984598, −5.09808561188218344168370245259, −4.22431401813120300732534441531, −3.74797915081579832167505600651, −2.72725899636677242130270388364, −1.53534522516950022267575400039, −0.31281065433610855304088904117, 0.31281065433610855304088904117, 1.53534522516950022267575400039, 2.72725899636677242130270388364, 3.74797915081579832167505600651, 4.22431401813120300732534441531, 5.09808561188218344168370245259, 5.59535575758865086949932984598, 6.44792033160151507544490550853, 7.05238747727061286609432930959, 7.81504786353188156896870995315

Graph of the ZZ-function along the critical line