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Properties

Label 2-150-5.4-c7-0-9
Degree 22
Conductor 150150
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 46.857746.8577
Root an. cond. 6.845276.84527
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  − 8i·2-s − 27i·3-s − 64·4-s − 216·6-s + 988i·7-s + 512i·8-s − 729·9-s − 8.04e3·11-s + 1.72e3i·12-s − 3.33e3i·13-s + 7.90e3·14-s + 4.09e3·16-s − 6.58e3i·17-s + 5.83e3i·18-s + 2.74e4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.08i·7-s + 0.353i·8-s − 0.333·9-s − 1.82·11-s + 0.288i·12-s − 0.420i·13-s + 0.769·14-s + 0.250·16-s − 0.324i·17-s + 0.235i·18-s + 0.917·19-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+7/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 46.857746.8577
Root analytic conductor: 6.845276.84527
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ150(49,⋅)\chi_{150} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :7/2), 0.447+0.894i)(2,\ 150,\ (\ :7/2),\ 0.447 + 0.894i)

Particular Values

L(4)L(4) ≈\approx 1.5218205461.521820546
L(12)L(\frac12) ≈\approx 1.5218205461.521820546
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+8iT 1 + 8iT
3 1+27iT 1 + 27iT
5 1 1
good7 1−988iT−8.23e5T2 1 - 988iT - 8.23e5T^{2}
11 1+8.04e3T+1.94e7T2 1 + 8.04e3T + 1.94e7T^{2}
13 1+3.33e3iT−6.27e7T2 1 + 3.33e3iT - 6.27e7T^{2}
17 1+6.58e3iT−4.10e8T2 1 + 6.58e3iT - 4.10e8T^{2}
19 1−2.74e4T+8.93e8T2 1 - 2.74e4T + 8.93e8T^{2}
23 1−4.86e4iT−3.40e9T2 1 - 4.86e4iT - 3.40e9T^{2}
29 1−1.32e5T+1.72e10T2 1 - 1.32e5T + 1.72e10T^{2}
31 1−2.54e5T+2.75e10T2 1 - 2.54e5T + 2.75e10T^{2}
37 1+5.19e5iT−9.49e10T2 1 + 5.19e5iT - 9.49e10T^{2}
41 1−9.23e4T+1.94e11T2 1 - 9.23e4T + 1.94e11T^{2}
43 1+2.34e5iT−2.71e11T2 1 + 2.34e5iT - 2.71e11T^{2}
47 1−1.27e6iT−5.06e11T2 1 - 1.27e6iT - 5.06e11T^{2}
53 1+8.35e5iT−1.17e12T2 1 + 8.35e5iT - 1.17e12T^{2}
59 1−3.06e6T+2.48e12T2 1 - 3.06e6T + 2.48e12T^{2}
61 1+1.00e6T+3.14e12T2 1 + 1.00e6T + 3.14e12T^{2}
67 1+3.08e6iT−6.06e12T2 1 + 3.08e6iT - 6.06e12T^{2}
71 1+3.66e6T+9.09e12T2 1 + 3.66e6T + 9.09e12T^{2}
73 1−1.12e6iT−1.10e13T2 1 - 1.12e6iT - 1.10e13T^{2}
79 1−4.12e6T+1.92e13T2 1 - 4.12e6T + 1.92e13T^{2}
83 1−4.58e6iT−2.71e13T2 1 - 4.58e6iT - 2.71e13T^{2}
89 1−5.76e6T+4.42e13T2 1 - 5.76e6T + 4.42e13T^{2}
97 1+6.74e6iT−8.07e13T2 1 + 6.74e6iT - 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

−11.66950331044554007293304700140, −10.59811131811901429942772406552, −9.551958405922810777014205554332, −8.378154957688452958374154989317, −7.54266166111324126413640217316, −5.81936462114732489458473556228, −5.02039526231186081966914078584, −3.02409600981238181618559015111, −2.29608609106302129126504774458, −0.68633964217214723757157626426, 0.69448766416874126982960865037, 2.89242788396088399298123728289, 4.34175353228633845951893783581, 5.19367120382675556141997339137, 6.57713362165829208589477584732, 7.70981577926883376693814812200, 8.535409816091331914409079973231, 10.11963302022571021446296684233, 10.37157253146118047905143806682, 11.82957637346468736596316953398

Graph of the ZZ-function along the critical line