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Properties

Label 2-252-9.4-c7-0-26
Degree 22
Conductor 252252
Sign 0.256+0.966i0.256 + 0.966i
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  + (−39.4 + 25.0i)3-s + (−137. + 238. i)5-s + (171.5 + 297. i)7-s + (928. − 1.98e3i)9-s + (1.72e3 + 2.99e3i)11-s + (1.83e3 − 3.17e3i)13-s + (−548. − 1.28e4i)15-s − 4.87e3·17-s − 4.01e4·19-s + (−1.42e4 − 7.42e3i)21-s + (−5.24e4 + 9.08e4i)23-s + (1.05e3 + 1.81e3i)25-s + (1.30e4 + 1.01e5i)27-s + (−1.59e4 − 2.76e4i)29-s + (−222. + 384. i)31-s + ⋯
L(s)  = 1  + (−0.843 + 0.536i)3-s + (−0.493 + 0.854i)5-s + (0.188 + 0.327i)7-s + (0.424 − 0.905i)9-s + (0.391 + 0.678i)11-s + (0.231 − 0.400i)13-s + (−0.0420 − 0.985i)15-s − 0.240·17-s − 1.34·19-s + (−0.335 − 0.174i)21-s + (−0.898 + 1.55i)23-s + (0.0134 + 0.0232i)25-s + (0.127 + 0.991i)27-s + (−0.121 − 0.210i)29-s + (−0.00133 + 0.00231i)31-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((0.256+0.966i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=((0.256+0.966i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 0.256+0.966i0.256 + 0.966i
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ252(85,⋅)\chi_{252} (85, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :7/2), 0.256+0.966i)(2,\ 252,\ (\ :7/2),\ 0.256 + 0.966i)

Particular Values

L(4)L(4) ≈\approx 0.079669334000.07966933400
L(12)L(\frac12) ≈\approx 0.079669334000.07966933400
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+(39.4−25.0i)T 1 + (39.4 - 25.0i)T
7 1+(−171.5−297.i)T 1 + (-171.5 - 297. i)T
good5 1+(137.−238.i)T+(−3.90e4−6.76e4i)T2 1 + (137. - 238. i)T + (-3.90e4 - 6.76e4i)T^{2}
11 1+(−1.72e3−2.99e3i)T+(−9.74e6+1.68e7i)T2 1 + (-1.72e3 - 2.99e3i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1+(−1.83e3+3.17e3i)T+(−3.13e7−5.43e7i)T2 1 + (-1.83e3 + 3.17e3i)T + (-3.13e7 - 5.43e7i)T^{2}
17 1+4.87e3T+4.10e8T2 1 + 4.87e3T + 4.10e8T^{2}
19 1+4.01e4T+8.93e8T2 1 + 4.01e4T + 8.93e8T^{2}
23 1+(5.24e4−9.08e4i)T+(−1.70e9−2.94e9i)T2 1 + (5.24e4 - 9.08e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(1.59e4+2.76e4i)T+(−8.62e9+1.49e10i)T2 1 + (1.59e4 + 2.76e4i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+(222.−384.i)T+(−1.37e10−2.38e10i)T2 1 + (222. - 384. i)T + (-1.37e10 - 2.38e10i)T^{2}
37 1+3.55e5T+9.49e10T2 1 + 3.55e5T + 9.49e10T^{2}
41 1+(1.25e5−2.17e5i)T+(−9.73e10−1.68e11i)T2 1 + (1.25e5 - 2.17e5i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+(6.08e4+1.05e5i)T+(−1.35e11+2.35e11i)T2 1 + (6.08e4 + 1.05e5i)T + (-1.35e11 + 2.35e11i)T^{2}
47 1+(1.61e5+2.78e5i)T+(−2.53e11+4.38e11i)T2 1 + (1.61e5 + 2.78e5i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1−1.10e6T+1.17e12T2 1 - 1.10e6T + 1.17e12T^{2}
59 1+(4.01e5−6.94e5i)T+(−1.24e12−2.15e12i)T2 1 + (4.01e5 - 6.94e5i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(1.68e5+2.92e5i)T+(−1.57e12+2.72e12i)T2 1 + (1.68e5 + 2.92e5i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(2.45e5−4.25e5i)T+(−3.03e12−5.24e12i)T2 1 + (2.45e5 - 4.25e5i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1+2.10e5T+9.09e12T2 1 + 2.10e5T + 9.09e12T^{2}
73 1+2.64e5T+1.10e13T2 1 + 2.64e5T + 1.10e13T^{2}
79 1+(2.08e5+3.60e5i)T+(−9.60e12+1.66e13i)T2 1 + (2.08e5 + 3.60e5i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(3.34e6+5.79e6i)T+(−1.35e13+2.35e13i)T2 1 + (3.34e6 + 5.79e6i)T + (-1.35e13 + 2.35e13i)T^{2}
89 1−6.64e6T+4.42e13T2 1 - 6.64e6T + 4.42e13T^{2}
97 1+(5.90e6+1.02e7i)T+(−4.03e13+6.99e13i)T2 1 + (5.90e6 + 1.02e7i)T + (-4.03e13 + 6.99e13i)T^{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

−10.65702140395782018591922368924, −9.906055995298427208656207600057, −8.756756774853994531098556589116, −7.43972104191848646122202175651, −6.54331345192000885567566815772, −5.55056300704063466889326481925, −4.32490064651485380447348885323, −3.40092678268323969975506648788, −1.77172188107172607892893826015, −0.02742580741355789381062988460, 0.821811367684440986932150709961, 2.03034825360359359578204126497, 4.02287317812146503441470280682, 4.80150049107905909892487241282, 6.08196242511453800933177920286, 6.87802079321306946935173072094, 8.195583356507811150426030250695, 8.748825666049932859383291382380, 10.34611496972009407630972375352, 11.07514504255465566627734938443

Graph of the ZZ-function along the critical line