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Properties

Label 2-252-21.20-c7-0-3
Degree 22
Conductor 252252
Sign −0.558−0.829i-0.558 - 0.829i
Analytic cond. 78.721078.7210
Root an. cond. 8.872488.87248
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 438.·5-s + (−907. − 20.5i)7-s − 7.41e3i·11-s + 2.70e3i·13-s − 3.34e4·17-s + 4.84e4i·19-s − 4.66e4i·23-s + 1.14e5·25-s + 1.59e5i·29-s + 2.14e5i·31-s + (−3.97e5 − 9.02e3i)35-s − 3.94e5·37-s + 1.57e5·41-s − 6.53e5·43-s + 9.95e5·47-s + ⋯
L(s)  = 1  + 1.56·5-s + (−0.999 − 0.0226i)7-s − 1.67i·11-s + 0.341i·13-s − 1.65·17-s + 1.62i·19-s − 0.799i·23-s + 1.45·25-s + 1.21i·29-s + 1.29i·31-s + (−1.56 − 0.0355i)35-s − 1.27·37-s + 0.357·41-s − 1.25·43-s + 1.39·47-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((−0.558−0.829i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=((−0.558−0.829i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: −0.558−0.829i-0.558 - 0.829i
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ252(125,⋅)\chi_{252} (125, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :7/2), −0.558−0.829i)(2,\ 252,\ (\ :7/2),\ -0.558 - 0.829i)

Particular Values

L(4)L(4) ≈\approx 0.96080309880.9608030988
L(12)L(\frac12) ≈\approx 0.96080309880.9608030988
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+(907.+20.5i)T 1 + (907. + 20.5i)T
good5 1−438.T+7.81e4T2 1 - 438.T + 7.81e4T^{2}
11 1+7.41e3iT−1.94e7T2 1 + 7.41e3iT - 1.94e7T^{2}
13 1−2.70e3iT−6.27e7T2 1 - 2.70e3iT - 6.27e7T^{2}
17 1+3.34e4T+4.10e8T2 1 + 3.34e4T + 4.10e8T^{2}
19 1−4.84e4iT−8.93e8T2 1 - 4.84e4iT - 8.93e8T^{2}
23 1+4.66e4iT−3.40e9T2 1 + 4.66e4iT - 3.40e9T^{2}
29 1−1.59e5iT−1.72e10T2 1 - 1.59e5iT - 1.72e10T^{2}
31 1−2.14e5iT−2.75e10T2 1 - 2.14e5iT - 2.75e10T^{2}
37 1+3.94e5T+9.49e10T2 1 + 3.94e5T + 9.49e10T^{2}
41 1−1.57e5T+1.94e11T2 1 - 1.57e5T + 1.94e11T^{2}
43 1+6.53e5T+2.71e11T2 1 + 6.53e5T + 2.71e11T^{2}
47 1−9.95e5T+5.06e11T2 1 - 9.95e5T + 5.06e11T^{2}
53 1−9.18e5iT−1.17e12T2 1 - 9.18e5iT - 1.17e12T^{2}
59 1−4.26e5T+2.48e12T2 1 - 4.26e5T + 2.48e12T^{2}
61 1−4.09e5iT−3.14e12T2 1 - 4.09e5iT - 3.14e12T^{2}
67 1−6.18e5T+6.06e12T2 1 - 6.18e5T + 6.06e12T^{2}
71 1+3.96e6iT−9.09e12T2 1 + 3.96e6iT - 9.09e12T^{2}
73 1−3.84e6iT−1.10e13T2 1 - 3.84e6iT - 1.10e13T^{2}
79 1+1.41e6T+1.92e13T2 1 + 1.41e6T + 1.92e13T^{2}
83 1−3.32e6T+2.71e13T2 1 - 3.32e6T + 2.71e13T^{2}
89 1+1.18e7T+4.42e13T2 1 + 1.18e7T + 4.42e13T^{2}
97 1−1.26e7iT−8.07e13T2 1 - 1.26e7iT - 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

−10.75649904581888820210865354631, −10.29590912870465145054754325396, −9.085129835687879166528814431339, −8.637232692106656838919730366963, −6.75051752132145359754431135235, −6.19580213740590127192137699882, −5.30987707100000370851901280260, −3.63887114526902902351473528636, −2.53259645265121289728780573518, −1.31611467938779697984593686586, 0.20154120910808619819896072550, 1.94212912626462876108530861554, 2.59298238727386906038897372949, 4.33655428389677930146089291116, 5.44893592809211345515949391257, 6.53081491178769414621409027345, 7.14758627147662990054967178536, 8.913315346341781743805726879095, 9.630629580802005690957911621525, 10.11683207545435088339161141081

Graph of the ZZ-function along the critical line