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Properties

Label 2-252-1.1-c7-0-5
Degree 22
Conductor 252252
Sign 11
Analytic cond. 78.721078.7210
Root an. cond. 8.872488.87248
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 166.·5-s + 343·7-s − 7.80e3·11-s + 1.06e4·13-s + 7.16e3·17-s − 1.95e4·19-s + 9.62e4·23-s − 5.03e4·25-s + 5.32e4·29-s + 2.85e5·31-s + 5.71e4·35-s − 3.10e5·37-s − 6.41e5·41-s + 2.18e5·43-s + 3.94e5·47-s + 1.17e5·49-s − 1.77e6·53-s − 1.29e6·55-s + 2.93e6·59-s + 2.16e6·61-s + 1.78e6·65-s + 4.35e6·67-s − 5.14e6·71-s + 3.57e6·73-s − 2.67e6·77-s − 4.59e6·79-s + 2.98e6·83-s + ⋯
L(s)  = 1  + 0.595·5-s + 0.377·7-s − 1.76·11-s + 1.34·13-s + 0.353·17-s − 0.653·19-s + 1.64·23-s − 0.644·25-s + 0.405·29-s + 1.72·31-s + 0.225·35-s − 1.00·37-s − 1.45·41-s + 0.419·43-s + 0.554·47-s + 0.142·49-s − 1.63·53-s − 1.05·55-s + 1.86·59-s + 1.21·61-s + 0.804·65-s + 1.76·67-s − 1.70·71-s + 1.07·73-s − 0.668·77-s − 1.04·79-s + 0.572·83-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 11
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 252, ( :7/2), 1)(2,\ 252,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 2.4600497212.460049721
L(12)L(\frac12) ≈\approx 2.4600497212.460049721
L(92)L(\frac{9}{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
7 1−343T 1 - 343T
good5 1−166.T+7.81e4T2 1 - 166.T + 7.81e4T^{2}
11 1+7.80e3T+1.94e7T2 1 + 7.80e3T + 1.94e7T^{2}
13 1−1.06e4T+6.27e7T2 1 - 1.06e4T + 6.27e7T^{2}
17 1−7.16e3T+4.10e8T2 1 - 7.16e3T + 4.10e8T^{2}
19 1+1.95e4T+8.93e8T2 1 + 1.95e4T + 8.93e8T^{2}
23 1−9.62e4T+3.40e9T2 1 - 9.62e4T + 3.40e9T^{2}
29 1−5.32e4T+1.72e10T2 1 - 5.32e4T + 1.72e10T^{2}
31 1−2.85e5T+2.75e10T2 1 - 2.85e5T + 2.75e10T^{2}
37 1+3.10e5T+9.49e10T2 1 + 3.10e5T + 9.49e10T^{2}
41 1+6.41e5T+1.94e11T2 1 + 6.41e5T + 1.94e11T^{2}
43 1−2.18e5T+2.71e11T2 1 - 2.18e5T + 2.71e11T^{2}
47 1−3.94e5T+5.06e11T2 1 - 3.94e5T + 5.06e11T^{2}
53 1+1.77e6T+1.17e12T2 1 + 1.77e6T + 1.17e12T^{2}
59 1−2.93e6T+2.48e12T2 1 - 2.93e6T + 2.48e12T^{2}
61 1−2.16e6T+3.14e12T2 1 - 2.16e6T + 3.14e12T^{2}
67 1−4.35e6T+6.06e12T2 1 - 4.35e6T + 6.06e12T^{2}
71 1+5.14e6T+9.09e12T2 1 + 5.14e6T + 9.09e12T^{2}
73 1−3.57e6T+1.10e13T2 1 - 3.57e6T + 1.10e13T^{2}
79 1+4.59e6T+1.92e13T2 1 + 4.59e6T + 1.92e13T^{2}
83 1−2.98e6T+2.71e13T2 1 - 2.98e6T + 2.71e13T^{2}
89 1−2.90e6T+4.42e13T2 1 - 2.90e6T + 4.42e13T^{2}
97 1−1.21e7T+8.07e13T2 1 - 1.21e7T + 8.07e13T^{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

−10.65819155830517763912234262883, −10.06571837371473898083876962866, −8.692088463505091335344011255136, −8.051543659249050474528227771095, −6.75393074847147578465765611811, −5.65493113346349105300938562989, −4.81292648738004275099533490883, −3.26522037853415610984067844734, −2.11686129491500488310944246942, −0.806203463818820225274944534999, 0.806203463818820225274944534999, 2.11686129491500488310944246942, 3.26522037853415610984067844734, 4.81292648738004275099533490883, 5.65493113346349105300938562989, 6.75393074847147578465765611811, 8.051543659249050474528227771095, 8.692088463505091335344011255136, 10.06571837371473898083876962866, 10.65819155830517763912234262883

Graph of the ZZ-function along the critical line