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Properties

Label 2-252-21.20-c7-0-18
Degree 22
Conductor 252252
Sign −0.452−0.891i-0.452 - 0.891i
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  − 524.·5-s + (423. − 802. i)7-s − 4.37e3i·11-s − 913. i·13-s − 3.44e4·17-s − 2.34e3i·19-s − 8.24e4i·23-s + 1.96e5·25-s − 2.26e5i·29-s − 1.42e5i·31-s + (−2.21e5 + 4.20e5i)35-s + 1.95e5·37-s + 3.77e5·41-s − 3.85e5·43-s + 1.39e5·47-s + ⋯
L(s)  = 1  − 1.87·5-s + (0.466 − 0.884i)7-s − 0.991i·11-s − 0.115i·13-s − 1.70·17-s − 0.0784i·19-s − 1.41i·23-s + 2.51·25-s − 1.72i·29-s − 0.858i·31-s + (−0.874 + 1.65i)35-s + 0.635·37-s + 0.855·41-s − 0.739·43-s + 0.196·47-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((−0.452−0.891i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=((−0.452−0.891i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: −0.452−0.891i-0.452 - 0.891i
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), −0.452−0.891i)(2,\ 252,\ (\ :7/2),\ -0.452 - 0.891i)

Particular Values

L(4)L(4) ≈\approx 0.22947520470.2294752047
L(12)L(\frac12) ≈\approx 0.22947520470.2294752047
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+(−423.+802.i)T 1 + (-423. + 802. i)T
good5 1+524.T+7.81e4T2 1 + 524.T + 7.81e4T^{2}
11 1+4.37e3iT−1.94e7T2 1 + 4.37e3iT - 1.94e7T^{2}
13 1+913.iT−6.27e7T2 1 + 913. iT - 6.27e7T^{2}
17 1+3.44e4T+4.10e8T2 1 + 3.44e4T + 4.10e8T^{2}
19 1+2.34e3iT−8.93e8T2 1 + 2.34e3iT - 8.93e8T^{2}
23 1+8.24e4iT−3.40e9T2 1 + 8.24e4iT - 3.40e9T^{2}
29 1+2.26e5iT−1.72e10T2 1 + 2.26e5iT - 1.72e10T^{2}
31 1+1.42e5iT−2.75e10T2 1 + 1.42e5iT - 2.75e10T^{2}
37 1−1.95e5T+9.49e10T2 1 - 1.95e5T + 9.49e10T^{2}
41 1−3.77e5T+1.94e11T2 1 - 3.77e5T + 1.94e11T^{2}
43 1+3.85e5T+2.71e11T2 1 + 3.85e5T + 2.71e11T^{2}
47 1−1.39e5T+5.06e11T2 1 - 1.39e5T + 5.06e11T^{2}
53 1−1.17e6iT−1.17e12T2 1 - 1.17e6iT - 1.17e12T^{2}
59 1+1.12e6T+2.48e12T2 1 + 1.12e6T + 2.48e12T^{2}
61 1−2.34e6iT−3.14e12T2 1 - 2.34e6iT - 3.14e12T^{2}
67 1+3.98e6T+6.06e12T2 1 + 3.98e6T + 6.06e12T^{2}
71 1−1.47e6iT−9.09e12T2 1 - 1.47e6iT - 9.09e12T^{2}
73 1+5.03e5iT−1.10e13T2 1 + 5.03e5iT - 1.10e13T^{2}
79 1−2.17e6T+1.92e13T2 1 - 2.17e6T + 1.92e13T^{2}
83 1+6.61e6T+2.71e13T2 1 + 6.61e6T + 2.71e13T^{2}
89 1+2.89e6T+4.42e13T2 1 + 2.89e6T + 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

−10.52188214093100618963089956257, −8.853610756580405001842778324827, −8.112225180139344368251681539883, −7.38392547790022344080859263414, −6.29199795793240571797876321015, −4.41477274177139218009217193164, −4.15273778258869236701836497078, −2.73608071941221802311330480808, −0.74039702956566711316416782765, −0.079482422362205239951931325900, 1.66587912471335502388619566248, 3.11685378132650504650180558273, 4.30428850204951432325551941047, 5.06337944965587269155087616793, 6.76614628298472115063137972867, 7.56165628172861858570162205067, 8.498174078304500130371471666739, 9.259593725824863707673615326742, 10.85467909924778245357848287143, 11.46406058447157839804725353773

Graph of the ZZ-function along the critical line