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Properties

Label 2-252-21.5-c7-0-13
Degree 22
Conductor 252252
Sign 0.341+0.939i0.341 + 0.939i
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  + (21.6 − 37.4i)5-s + (487. + 765. i)7-s + (2.46e3 − 1.42e3i)11-s − 1.90e3i·13-s + (−1.71e4 − 2.97e4i)17-s + (−2.75e4 − 1.59e4i)19-s + (3.58e4 + 2.06e4i)23-s + (3.81e4 + 6.60e4i)25-s − 3.99e3i·29-s + (−1.10e5 + 6.39e4i)31-s + (3.92e4 − 1.70e3i)35-s + (2.43e5 − 4.21e5i)37-s − 1.19e4·41-s + 3.23e5·43-s + (−4.71e5 + 8.17e5i)47-s + ⋯
L(s)  = 1  + (0.0774 − 0.134i)5-s + (0.537 + 0.843i)7-s + (0.558 − 0.322i)11-s − 0.240i·13-s + (−0.848 − 1.46i)17-s + (−0.923 − 0.532i)19-s + (0.613 + 0.354i)23-s + (0.488 + 0.845i)25-s − 0.0304i·29-s + (−0.668 + 0.385i)31-s + (0.154 − 0.00673i)35-s + (0.790 − 1.36i)37-s − 0.0271·41-s + 0.619·43-s + (−0.662 + 1.14i)47-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((0.341+0.939i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=((0.341+0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 0.341+0.939i0.341 + 0.939i
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ252(89,⋅)\chi_{252} (89, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :7/2), 0.341+0.939i)(2,\ 252,\ (\ :7/2),\ 0.341 + 0.939i)

Particular Values

L(4)L(4) ≈\approx 1.8792986421.879298642
L(12)L(\frac12) ≈\approx 1.8792986421.879298642
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+(−487.−765.i)T 1 + (-487. - 765. i)T
good5 1+(−21.6+37.4i)T+(−3.90e4−6.76e4i)T2 1 + (-21.6 + 37.4i)T + (-3.90e4 - 6.76e4i)T^{2}
11 1+(−2.46e3+1.42e3i)T+(9.74e6−1.68e7i)T2 1 + (-2.46e3 + 1.42e3i)T + (9.74e6 - 1.68e7i)T^{2}
13 1+1.90e3iT−6.27e7T2 1 + 1.90e3iT - 6.27e7T^{2}
17 1+(1.71e4+2.97e4i)T+(−2.05e8+3.55e8i)T2 1 + (1.71e4 + 2.97e4i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(2.75e4+1.59e4i)T+(4.46e8+7.74e8i)T2 1 + (2.75e4 + 1.59e4i)T + (4.46e8 + 7.74e8i)T^{2}
23 1+(−3.58e4−2.06e4i)T+(1.70e9+2.94e9i)T2 1 + (-3.58e4 - 2.06e4i)T + (1.70e9 + 2.94e9i)T^{2}
29 1+3.99e3iT−1.72e10T2 1 + 3.99e3iT - 1.72e10T^{2}
31 1+(1.10e5−6.39e4i)T+(1.37e10−2.38e10i)T2 1 + (1.10e5 - 6.39e4i)T + (1.37e10 - 2.38e10i)T^{2}
37 1+(−2.43e5+4.21e5i)T+(−4.74e10−8.22e10i)T2 1 + (-2.43e5 + 4.21e5i)T + (-4.74e10 - 8.22e10i)T^{2}
41 1+1.19e4T+1.94e11T2 1 + 1.19e4T + 1.94e11T^{2}
43 1−3.23e5T+2.71e11T2 1 - 3.23e5T + 2.71e11T^{2}
47 1+(4.71e5−8.17e5i)T+(−2.53e11−4.38e11i)T2 1 + (4.71e5 - 8.17e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+(−1.78e6+1.02e6i)T+(5.87e11−1.01e12i)T2 1 + (-1.78e6 + 1.02e6i)T + (5.87e11 - 1.01e12i)T^{2}
59 1+(5.58e5+9.66e5i)T+(−1.24e12+2.15e12i)T2 1 + (5.58e5 + 9.66e5i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(2.64e6+1.52e6i)T+(1.57e12+2.72e12i)T2 1 + (2.64e6 + 1.52e6i)T + (1.57e12 + 2.72e12i)T^{2}
67 1+(5.02e5+8.70e5i)T+(−3.03e12+5.24e12i)T2 1 + (5.02e5 + 8.70e5i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+2.34e6iT−9.09e12T2 1 + 2.34e6iT - 9.09e12T^{2}
73 1+(−1.94e6+1.12e6i)T+(5.52e12−9.56e12i)T2 1 + (-1.94e6 + 1.12e6i)T + (5.52e12 - 9.56e12i)T^{2}
79 1+(4.54e5−7.87e5i)T+(−9.60e12−1.66e13i)T2 1 + (4.54e5 - 7.87e5i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1−4.64e6T+2.71e13T2 1 - 4.64e6T + 2.71e13T^{2}
89 1+(−4.97e6+8.61e6i)T+(−2.21e13−3.83e13i)T2 1 + (-4.97e6 + 8.61e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1+7.88e6iT−8.07e13T2 1 + 7.88e6iT - 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

−10.92415942631459992347089365712, −9.201985165235257596839476917142, −9.021284030162887655550099469961, −7.67584996562799836825615509789, −6.60300099640709277067383961371, −5.43873969106213135240888124938, −4.54993009567162287901109573810, −3.02697648887917276743922927789, −1.89801265151282306826895420482, −0.47064611695530770147848396880, 1.10166634157769611502831489800, 2.20446374423611244355238936414, 3.89924038352375647600232836076, 4.57084974631058641789185397313, 6.14955259945402330685263142052, 6.94517567960920581144319230097, 8.118176022043949140745005204405, 8.944307107173077261651582751497, 10.29650460531866653824023902663, 10.78810600702103762893257292968

Graph of the ZZ-function along the critical line