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Properties

Label 2-8512-1.1-c1-0-25
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  + 0.811·3-s − 0.636·5-s − 7-s − 2.34·9-s − 1.29·11-s − 1.02·13-s − 0.516·15-s − 7.09·17-s + 19-s − 0.811·21-s − 3.70·23-s − 4.59·25-s − 4.33·27-s − 4.94·29-s + 2.96·31-s − 1.05·33-s + 0.636·35-s − 2.78·37-s − 0.828·39-s + 5.64·41-s − 1.08·43-s + 1.48·45-s + 11.2·47-s + 49-s − 5.75·51-s + 6.23·53-s + 0.825·55-s + ⋯
L(s)  = 1  + 0.468·3-s − 0.284·5-s − 0.377·7-s − 0.780·9-s − 0.391·11-s − 0.283·13-s − 0.133·15-s − 1.72·17-s + 0.229·19-s − 0.177·21-s − 0.771·23-s − 0.919·25-s − 0.834·27-s − 0.918·29-s + 0.532·31-s − 0.183·33-s + 0.107·35-s − 0.457·37-s − 0.132·39-s + 0.881·41-s − 0.165·43-s + 0.221·45-s + 1.63·47-s + 0.142·49-s − 0.806·51-s + 0.857·53-s + 0.111·55-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 1.0336219241.033621924
L(12)L(\frac12) ≈\approx 1.0336219241.033621924
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 1−0.811T+3T2 1 - 0.811T + 3T^{2}
5 1+0.636T+5T2 1 + 0.636T + 5T^{2}
11 1+1.29T+11T2 1 + 1.29T + 11T^{2}
13 1+1.02T+13T2 1 + 1.02T + 13T^{2}
17 1+7.09T+17T2 1 + 7.09T + 17T^{2}
23 1+3.70T+23T2 1 + 3.70T + 23T^{2}
29 1+4.94T+29T2 1 + 4.94T + 29T^{2}
31 1−2.96T+31T2 1 - 2.96T + 31T^{2}
37 1+2.78T+37T2 1 + 2.78T + 37T^{2}
41 1−5.64T+41T2 1 - 5.64T + 41T^{2}
43 1+1.08T+43T2 1 + 1.08T + 43T^{2}
47 1−11.2T+47T2 1 - 11.2T + 47T^{2}
53 1−6.23T+53T2 1 - 6.23T + 53T^{2}
59 1+1.27T+59T2 1 + 1.27T + 59T^{2}
61 1−11.3T+61T2 1 - 11.3T + 61T^{2}
67 1+7.11T+67T2 1 + 7.11T + 67T^{2}
71 1−4.45T+71T2 1 - 4.45T + 71T^{2}
73 1+2.12T+73T2 1 + 2.12T + 73T^{2}
79 1−13.1T+79T2 1 - 13.1T + 79T^{2}
83 1+2.08T+83T2 1 + 2.08T + 83T^{2}
89 1−7.92T+89T2 1 - 7.92T + 89T^{2}
97 1−9.66T+97T2 1 - 9.66T + 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

−7.72961778662445013636968132176, −7.28824155661925908715011318098, −6.34073426624168775997540125686, −5.81139121391015738485781469393, −4.98463744833399596467434061899, −4.08362343646535925969837915930, −3.55130060072728447283973844501, −2.47727663257257431154858621512, −2.13789277956346894997400320308, −0.44959341818561389998374778731, 0.44959341818561389998374778731, 2.13789277956346894997400320308, 2.47727663257257431154858621512, 3.55130060072728447283973844501, 4.08362343646535925969837915930, 4.98463744833399596467434061899, 5.81139121391015738485781469393, 6.34073426624168775997540125686, 7.28824155661925908715011318098, 7.72961778662445013636968132176

Graph of the ZZ-function along the critical line