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Properties

Label 2-252-21.20-c7-0-15
Degree 22
Conductor 252252
Sign 0.0686+0.997i0.0686 + 0.997i
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  + 209.·5-s + (775. + 471. i)7-s − 2.15e3i·11-s − 6.16e3i·13-s − 1.20e4·17-s − 3.52e4i·19-s − 3.86e4i·23-s − 3.43e4·25-s − 4.46e4i·29-s − 3.92e4i·31-s + (1.62e5 + 9.87e4i)35-s − 9.53e4·37-s + 3.59e5·41-s + 1.52e5·43-s − 5.22e5·47-s + ⋯
L(s)  = 1  + 0.748·5-s + (0.854 + 0.519i)7-s − 0.488i·11-s − 0.777i·13-s − 0.593·17-s − 1.17i·19-s − 0.662i·23-s − 0.439·25-s − 0.339i·29-s − 0.236i·31-s + (0.639 + 0.389i)35-s − 0.309·37-s + 0.813·41-s + 0.293·43-s − 0.733·47-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((0.0686+0.997i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0686 + 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=((0.0686+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0686 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 0.0686+0.997i0.0686 + 0.997i
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), 0.0686+0.997i)(2,\ 252,\ (\ :7/2),\ 0.0686 + 0.997i)

Particular Values

L(4)L(4) ≈\approx 2.1747943562.174794356
L(12)L(\frac12) ≈\approx 2.1747943562.174794356
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+(−775.−471.i)T 1 + (-775. - 471. i)T
good5 1−209.T+7.81e4T2 1 - 209.T + 7.81e4T^{2}
11 1+2.15e3iT−1.94e7T2 1 + 2.15e3iT - 1.94e7T^{2}
13 1+6.16e3iT−6.27e7T2 1 + 6.16e3iT - 6.27e7T^{2}
17 1+1.20e4T+4.10e8T2 1 + 1.20e4T + 4.10e8T^{2}
19 1+3.52e4iT−8.93e8T2 1 + 3.52e4iT - 8.93e8T^{2}
23 1+3.86e4iT−3.40e9T2 1 + 3.86e4iT - 3.40e9T^{2}
29 1+4.46e4iT−1.72e10T2 1 + 4.46e4iT - 1.72e10T^{2}
31 1+3.92e4iT−2.75e10T2 1 + 3.92e4iT - 2.75e10T^{2}
37 1+9.53e4T+9.49e10T2 1 + 9.53e4T + 9.49e10T^{2}
41 1−3.59e5T+1.94e11T2 1 - 3.59e5T + 1.94e11T^{2}
43 1−1.52e5T+2.71e11T2 1 - 1.52e5T + 2.71e11T^{2}
47 1+5.22e5T+5.06e11T2 1 + 5.22e5T + 5.06e11T^{2}
53 1+8.02e4iT−1.17e12T2 1 + 8.02e4iT - 1.17e12T^{2}
59 1+2.58e6T+2.48e12T2 1 + 2.58e6T + 2.48e12T^{2}
61 1+2.42e5iT−3.14e12T2 1 + 2.42e5iT - 3.14e12T^{2}
67 1−2.42e6T+6.06e12T2 1 - 2.42e6T + 6.06e12T^{2}
71 1+2.71e6iT−9.09e12T2 1 + 2.71e6iT - 9.09e12T^{2}
73 1+1.96e6iT−1.10e13T2 1 + 1.96e6iT - 1.10e13T^{2}
79 1−2.72e6T+1.92e13T2 1 - 2.72e6T + 1.92e13T^{2}
83 1+3.10e6T+2.71e13T2 1 + 3.10e6T + 2.71e13T^{2}
89 1−1.57e6T+4.42e13T2 1 - 1.57e6T + 4.42e13T^{2}
97 1+1.06e7iT−8.07e13T2 1 + 1.06e7iT - 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

−10.70006900928762988767395541660, −9.519427887094501159912216808131, −8.676322131839851044882467621048, −7.74747705310515996231219353253, −6.39601069146260982961240307912, −5.50113461702600591900789798765, −4.52433737397656033266824045436, −2.86625560717962111477970079477, −1.88298851425727505901775127964, −0.48084195043951146869959708830, 1.35003009209943599500158213893, 2.12085954080265360350546634410, 3.84431258512652518287223809086, 4.87291493184342804744747979667, 5.98702831758723675089565251693, 7.09895680525900502325844177509, 8.063528588288816653191919092967, 9.213165143363381201516719724228, 10.05237767451851644141757043791, 10.98543896256466747889938905207

Graph of the ZZ-function along the critical line