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Properties

Label 2-252-7.2-c5-0-11
Degree 22
Conductor 252252
Sign 0.415+0.909i0.415 + 0.909i
Analytic cond. 40.416740.4167
Root an. cond. 6.357416.35741
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (39.3 − 68.1i)5-s + (−100. − 81.7i)7-s + (345. + 598. i)11-s + 818.·13-s + (554. + 960. i)17-s + (286. − 496. i)19-s + (1.25e3 − 2.18e3i)23-s + (−1.53e3 − 2.65e3i)25-s + 3.25e3·29-s + (−5.05e3 − 8.76e3i)31-s + (−9.52e3 + 3.63e3i)35-s + (−2.43e3 + 4.21e3i)37-s + 1.30e4·41-s − 9.30e3·43-s + (6.45e3 − 1.11e4i)47-s + ⋯
L(s)  = 1  + (0.703 − 1.21i)5-s + (−0.776 − 0.630i)7-s + (0.861 + 1.49i)11-s + 1.34·13-s + (0.465 + 0.806i)17-s + (0.182 − 0.315i)19-s + (0.496 − 0.859i)23-s + (−0.490 − 0.849i)25-s + 0.719·29-s + (−0.945 − 1.63i)31-s + (−1.31 + 0.502i)35-s + (−0.292 + 0.506i)37-s + 1.21·41-s − 0.767·43-s + (0.426 − 0.738i)47-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((0.415+0.909i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+5/2) L(s)=((0.415+0.909i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 0.415+0.909i0.415 + 0.909i
Analytic conductor: 40.416740.4167
Root analytic conductor: 6.357416.35741
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ252(37,⋅)\chi_{252} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :5/2), 0.415+0.909i)(2,\ 252,\ (\ :5/2),\ 0.415 + 0.909i)

Particular Values

L(3)L(3) ≈\approx 2.3864326252.386432625
L(12)L(\frac12) ≈\approx 2.3864326252.386432625
L(72)L(\frac{7}{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+(100.+81.7i)T 1 + (100. + 81.7i)T
good5 1+(−39.3+68.1i)T+(−1.56e3−2.70e3i)T2 1 + (-39.3 + 68.1i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(−345.−598.i)T+(−8.05e4+1.39e5i)T2 1 + (-345. - 598. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−818.T+3.71e5T2 1 - 818.T + 3.71e5T^{2}
17 1+(−554.−960.i)T+(−7.09e5+1.22e6i)T2 1 + (-554. - 960. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−286.+496.i)T+(−1.23e6−2.14e6i)T2 1 + (-286. + 496. i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(−1.25e3+2.18e3i)T+(−3.21e6−5.57e6i)T2 1 + (-1.25e3 + 2.18e3i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−3.25e3T+2.05e7T2 1 - 3.25e3T + 2.05e7T^{2}
31 1+(5.05e3+8.76e3i)T+(−1.43e7+2.47e7i)T2 1 + (5.05e3 + 8.76e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(2.43e3−4.21e3i)T+(−3.46e7−6.00e7i)T2 1 + (2.43e3 - 4.21e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−1.30e4T+1.15e8T2 1 - 1.30e4T + 1.15e8T^{2}
43 1+9.30e3T+1.47e8T2 1 + 9.30e3T + 1.47e8T^{2}
47 1+(−6.45e3+1.11e4i)T+(−1.14e8−1.98e8i)T2 1 + (-6.45e3 + 1.11e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(9.77e3+1.69e4i)T+(−2.09e8+3.62e8i)T2 1 + (9.77e3 + 1.69e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(1.25e4+2.17e4i)T+(−3.57e8+6.19e8i)T2 1 + (1.25e4 + 2.17e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(−1.56e4+2.71e4i)T+(−4.22e8−7.31e8i)T2 1 + (-1.56e4 + 2.71e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(2.79e4+4.84e4i)T+(−6.75e8+1.16e9i)T2 1 + (2.79e4 + 4.84e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+2.05e4T+1.80e9T2 1 + 2.05e4T + 1.80e9T^{2}
73 1+(−3.38e4−5.85e4i)T+(−1.03e9+1.79e9i)T2 1 + (-3.38e4 - 5.85e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(7.03e3−1.21e4i)T+(−1.53e9−2.66e9i)T2 1 + (7.03e3 - 1.21e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1−7.71e4T+3.93e9T2 1 - 7.71e4T + 3.93e9T^{2}
89 1+(−160.+277.i)T+(−2.79e9−4.83e9i)T2 1 + (-160. + 277. i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1+1.12e5T+8.58e9T2 1 + 1.12e5T + 8.58e9T^{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

−10.90368383188837416111849991405, −9.798839583721829296705283508013, −9.275095535148949458221282211382, −8.229222368643994247372610068220, −6.85594497469227791264481988558, −5.99793395320257582284441348414, −4.68624483818697190503454745326, −3.74032992166119261071068876800, −1.81624284247605151347362328761, −0.797144217960365651707493987760, 1.20334725628650198460127730390, 2.94167917997169344042337746507, 3.49225454350231285716834257931, 5.69348724733628945330041042892, 6.18934049461295855798113830056, 7.14590706211169204850096361106, 8.710756197050332085817890656746, 9.356806169626800297242103306230, 10.53606112551797535715733134737, 11.19385779710262615470566224633

Graph of the ZZ-function along the critical line