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Properties

Label 2-51-3.2-c6-0-16
Degree 22
Conductor 5151
Sign 0.577−0.816i0.577 - 0.816i
Analytic cond. 11.732711.7327
Root an. cond. 3.425313.42531
Motivic weight 66
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.461i·2-s + (22.0 + 15.5i)3-s + 63.7·4-s + 70.0i·5-s + (−7.19 + 10.1i)6-s + 296.·7-s + 58.9i·8-s + (242. + 687. i)9-s − 32.3·10-s − 1.24e3i·11-s + (1.40e3 + 994. i)12-s − 2.89e3·13-s + 136. i·14-s + (−1.09e3 + 1.54e3i)15-s + 4.05e3·16-s + 1.19e3i·17-s + ⋯
L(s)  = 1  + 0.0576i·2-s + (0.816 + 0.577i)3-s + 0.996·4-s + 0.560i·5-s + (−0.0333 + 0.0470i)6-s + 0.864·7-s + 0.115i·8-s + (0.332 + 0.942i)9-s − 0.0323·10-s − 0.937i·11-s + (0.813 + 0.575i)12-s − 1.31·13-s + 0.0498i·14-s + (−0.323 + 0.457i)15-s + 0.990·16-s + 0.242i·17-s + ⋯

Functional equation

Λ(s)=(51s/2 ΓC(s) L(s)=((0.577−0.816i) Λ‾(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}
Λ(s)=(51s/2 ΓC(s+3) L(s)=((0.577−0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5151    =    3⋅173 \cdot 17
Sign: 0.577−0.816i0.577 - 0.816i
Analytic conductor: 11.732711.7327
Root analytic conductor: 3.425313.42531
Motivic weight: 66
Rational: no
Arithmetic: yes
Character: χ51(35,⋅)\chi_{51} (35, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 51, ( :3), 0.577−0.816i)(2,\ 51,\ (\ :3),\ 0.577 - 0.816i)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 2.64348+1.36795i2.64348 + 1.36795i
L(12)L(\frac12) ≈\approx 2.64348+1.36795i2.64348 + 1.36795i
L(4)L(4) 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)
bad3 1+(−22.0−15.5i)T 1 + (-22.0 - 15.5i)T
17 1−1.19e3iT 1 - 1.19e3iT
good2 1−0.461iT−64T2 1 - 0.461iT - 64T^{2}
5 1−70.0iT−1.56e4T2 1 - 70.0iT - 1.56e4T^{2}
7 1−296.T+1.17e5T2 1 - 296.T + 1.17e5T^{2}
11 1+1.24e3iT−1.77e6T2 1 + 1.24e3iT - 1.77e6T^{2}
13 1+2.89e3T+4.82e6T2 1 + 2.89e3T + 4.82e6T^{2}
19 1−3.45e3T+4.70e7T2 1 - 3.45e3T + 4.70e7T^{2}
23 1−2.09e4iT−1.48e8T2 1 - 2.09e4iT - 1.48e8T^{2}
29 1+4.38e4iT−5.94e8T2 1 + 4.38e4iT - 5.94e8T^{2}
31 1+1.56e4T+8.87e8T2 1 + 1.56e4T + 8.87e8T^{2}
37 1+5.20e4T+2.56e9T2 1 + 5.20e4T + 2.56e9T^{2}
41 1+1.35e4iT−4.75e9T2 1 + 1.35e4iT - 4.75e9T^{2}
43 1−1.91e4T+6.32e9T2 1 - 1.91e4T + 6.32e9T^{2}
47 1+1.38e5iT−1.07e10T2 1 + 1.38e5iT - 1.07e10T^{2}
53 1+2.59e5iT−2.21e10T2 1 + 2.59e5iT - 2.21e10T^{2}
59 1+2.61e4iT−4.21e10T2 1 + 2.61e4iT - 4.21e10T^{2}
61 1−1.15e5T+5.15e10T2 1 - 1.15e5T + 5.15e10T^{2}
67 1+4.47e5T+9.04e10T2 1 + 4.47e5T + 9.04e10T^{2}
71 1+8.44e4iT−1.28e11T2 1 + 8.44e4iT - 1.28e11T^{2}
73 1−3.29e4T+1.51e11T2 1 - 3.29e4T + 1.51e11T^{2}
79 1−8.37e5T+2.43e11T2 1 - 8.37e5T + 2.43e11T^{2}
83 1−4.85e5iT−3.26e11T2 1 - 4.85e5iT - 3.26e11T^{2}
89 1+2.78e5iT−4.96e11T2 1 + 2.78e5iT - 4.96e11T^{2}
97 1−2.31e5T+8.32e11T2 1 - 2.31e5T + 8.32e11T^{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

−14.67782434635763190526512310382, −13.64461224306223173115901814067, −11.79635836172090247396154042713, −10.90120763815226268871719158060, −9.775410742943872250245343119668, −8.148471115990954300941624667835, −7.21384085516388074033186526143, −5.34523826021561221711233637108, −3.38119574331882075064578193994, −2.05726524105390905381281587475, 1.42426031927839091925402624190, 2.65696729437365178757575479409, 4.83830615884208094933173334200, 6.90571620242803621333662865370, 7.73941540891718132872842799240, 9.090043118662467697412508970653, 10.55353553443497918496838900485, 12.16951625913858082315986246723, 12.57348602610952265396605405623, 14.40498488973305892497847252323

Graph of the ZZ-function along the critical line