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Properties

Label 2-150-15.2-c7-0-24
Degree 22
Conductor 150150
Sign 0.245+0.969i0.245 + 0.969i
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  + (−5.65 + 5.65i)2-s + (−0.750 − 46.7i)3-s − 64.0i·4-s + (268. + 260. i)6-s + (661. + 661. i)7-s + (362. + 362. i)8-s + (−2.18e3 + 70.1i)9-s − 6.67e3i·11-s + (−2.99e3 + 48.0i)12-s + (4.25e3 − 4.25e3i)13-s − 7.48e3·14-s − 4.09e3·16-s + (1.15e4 − 1.15e4i)17-s + (1.19e4 − 1.27e4i)18-s + 5.65e4i·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.0160 − 0.999i)3-s − 0.500i·4-s + (0.507 + 0.491i)6-s + (0.729 + 0.729i)7-s + (0.250 + 0.250i)8-s + (−0.999 + 0.0320i)9-s − 1.51i·11-s + (−0.499 + 0.00801i)12-s + (0.536 − 0.536i)13-s − 0.729·14-s − 0.250·16-s + (0.568 − 0.568i)17-s + (0.483 − 0.515i)18-s + 1.89i·19-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((0.245+0.969i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+7/2) L(s)=((0.245+0.969i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: 0.245+0.969i0.245 + 0.969i
Analytic conductor: 46.857746.8577
Root analytic conductor: 6.845276.84527
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ150(107,⋅)\chi_{150} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :7/2), 0.245+0.969i)(2,\ 150,\ (\ :7/2),\ 0.245 + 0.969i)

Particular Values

L(4)L(4) ≈\approx 1.5492859461.549285946
L(12)L(\frac12) ≈\approx 1.5492859461.549285946
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+(5.65−5.65i)T 1 + (5.65 - 5.65i)T
3 1+(0.750+46.7i)T 1 + (0.750 + 46.7i)T
5 1 1
good7 1+(−661.−661.i)T+8.23e5iT2 1 + (-661. - 661. i)T + 8.23e5iT^{2}
11 1+6.67e3iT−1.94e7T2 1 + 6.67e3iT - 1.94e7T^{2}
13 1+(−4.25e3+4.25e3i)T−6.27e7iT2 1 + (-4.25e3 + 4.25e3i)T - 6.27e7iT^{2}
17 1+(−1.15e4+1.15e4i)T−4.10e8iT2 1 + (-1.15e4 + 1.15e4i)T - 4.10e8iT^{2}
19 1−5.65e4iT−8.93e8T2 1 - 5.65e4iT - 8.93e8T^{2}
23 1+(−3.68e4−3.68e4i)T+3.40e9iT2 1 + (-3.68e4 - 3.68e4i)T + 3.40e9iT^{2}
29 1+3.43e4T+1.72e10T2 1 + 3.43e4T + 1.72e10T^{2}
31 1−1.54e5T+2.75e10T2 1 - 1.54e5T + 2.75e10T^{2}
37 1+(8.63e4+8.63e4i)T+9.49e10iT2 1 + (8.63e4 + 8.63e4i)T + 9.49e10iT^{2}
41 1+5.20e5iT−1.94e11T2 1 + 5.20e5iT - 1.94e11T^{2}
43 1+(−4.99e5+4.99e5i)T−2.71e11iT2 1 + (-4.99e5 + 4.99e5i)T - 2.71e11iT^{2}
47 1+(−4.71e4+4.71e4i)T−5.06e11iT2 1 + (-4.71e4 + 4.71e4i)T - 5.06e11iT^{2}
53 1+(−3.15e5−3.15e5i)T+1.17e12iT2 1 + (-3.15e5 - 3.15e5i)T + 1.17e12iT^{2}
59 1+1.04e6T+2.48e12T2 1 + 1.04e6T + 2.48e12T^{2}
61 1+1.20e6T+3.14e12T2 1 + 1.20e6T + 3.14e12T^{2}
67 1+(2.42e5+2.42e5i)T+6.06e12iT2 1 + (2.42e5 + 2.42e5i)T + 6.06e12iT^{2}
71 1−3.41e4iT−9.09e12T2 1 - 3.41e4iT - 9.09e12T^{2}
73 1+(−2.31e6+2.31e6i)T−1.10e13iT2 1 + (-2.31e6 + 2.31e6i)T - 1.10e13iT^{2}
79 1+7.82e6iT−1.92e13T2 1 + 7.82e6iT - 1.92e13T^{2}
83 1+(5.24e6+5.24e6i)T+2.71e13iT2 1 + (5.24e6 + 5.24e6i)T + 2.71e13iT^{2}
89 1+5.17e6T+4.42e13T2 1 + 5.17e6T + 4.42e13T^{2}
97 1+(1.14e5+1.14e5i)T+8.07e13iT2 1 + (1.14e5 + 1.14e5i)T + 8.07e13iT^{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

−11.52638420605878934528846721308, −10.55616746471011188494186071467, −8.935737582925310153412544214585, −8.262478201117426844660310286188, −7.47593409009046536804969387808, −5.93527514044533818341003350510, −5.55681491514889901975985187630, −3.22569408344256229939944122733, −1.66974650596095551923432082222, −0.59208310091006467160891965008, 1.11750271827958720691982010511, 2.65110813785814079154078554669, 4.20105228012598375199797384763, 4.82128532744342991870341888683, 6.74503483332429036310347206918, 7.962962844353988178905431578491, 9.079072094313401146570215503282, 9.917971612307814084834245401037, 10.84971524214486008778665797828, 11.47488889942030088558546180632

Graph of the ZZ-function along the critical line