↑

Properties

Label 2-150-15.2-c7-0-19
Degree 22
Conductor 150150
Sign 0.748+0.663i0.748 + 0.663i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (5.65 − 5.65i)2-s + (−38.2 + 26.9i)3-s − 64.0i·4-s + (−63.8 + 368. i)6-s + (−6.12 − 6.12i)7-s + (−362. − 362. i)8-s + (735. − 2.05e3i)9-s + 2.86e3i·11-s + (1.72e3 + 2.44e3i)12-s + (−2.01e3 + 2.01e3i)13-s − 69.3·14-s − 4.09e3·16-s + (−8.71e3 + 8.71e3i)17-s + (−7.49e3 − 1.58e4i)18-s − 1.40e3i·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.817 + 0.576i)3-s − 0.500i·4-s + (−0.120 + 0.696i)6-s + (−0.00675 − 0.00675i)7-s + (−0.250 − 0.250i)8-s + (0.336 − 0.941i)9-s + 0.648i·11-s + (0.288 + 0.408i)12-s + (−0.254 + 0.254i)13-s − 0.00675·14-s − 0.250·16-s + (−0.430 + 0.430i)17-s + (−0.302 − 0.639i)18-s − 0.0471i·19-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((0.748+0.663i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+7/2) L(s)=((0.748+0.663i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: 0.748+0.663i0.748 + 0.663i
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), 0.748+0.663i)(2,\ 150,\ (\ :7/2),\ 0.748 + 0.663i)

Particular Values

L(4)L(4) ≈\approx 1.7348919551.734891955
L(12)L(\frac12) ≈\approx 1.7348919551.734891955
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+(38.2−26.9i)T 1 + (38.2 - 26.9i)T
5 1 1
good7 1+(6.12+6.12i)T+8.23e5iT2 1 + (6.12 + 6.12i)T + 8.23e5iT^{2}
11 1−2.86e3iT−1.94e7T2 1 - 2.86e3iT - 1.94e7T^{2}
13 1+(2.01e3−2.01e3i)T−6.27e7iT2 1 + (2.01e3 - 2.01e3i)T - 6.27e7iT^{2}
17 1+(8.71e3−8.71e3i)T−4.10e8iT2 1 + (8.71e3 - 8.71e3i)T - 4.10e8iT^{2}
19 1+1.40e3iT−8.93e8T2 1 + 1.40e3iT - 8.93e8T^{2}
23 1+(5.24e4+5.24e4i)T+3.40e9iT2 1 + (5.24e4 + 5.24e4i)T + 3.40e9iT^{2}
29 1−9.58e4T+1.72e10T2 1 - 9.58e4T + 1.72e10T^{2}
31 1−1.25e5T+2.75e10T2 1 - 1.25e5T + 2.75e10T^{2}
37 1+(−3.05e5−3.05e5i)T+9.49e10iT2 1 + (-3.05e5 - 3.05e5i)T + 9.49e10iT^{2}
41 1+6.09e5iT−1.94e11T2 1 + 6.09e5iT - 1.94e11T^{2}
43 1+(−3.17e5+3.17e5i)T−2.71e11iT2 1 + (-3.17e5 + 3.17e5i)T - 2.71e11iT^{2}
47 1+(−4.35e5+4.35e5i)T−5.06e11iT2 1 + (-4.35e5 + 4.35e5i)T - 5.06e11iT^{2}
53 1+(−1.07e6−1.07e6i)T+1.17e12iT2 1 + (-1.07e6 - 1.07e6i)T + 1.17e12iT^{2}
59 1−3.09e6T+2.48e12T2 1 - 3.09e6T + 2.48e12T^{2}
61 1−1.62e6T+3.14e12T2 1 - 1.62e6T + 3.14e12T^{2}
67 1+(9.07e5+9.07e5i)T+6.06e12iT2 1 + (9.07e5 + 9.07e5i)T + 6.06e12iT^{2}
71 1+3.60e6iT−9.09e12T2 1 + 3.60e6iT - 9.09e12T^{2}
73 1+(2.84e6−2.84e6i)T−1.10e13iT2 1 + (2.84e6 - 2.84e6i)T - 1.10e13iT^{2}
79 1+4.14e6iT−1.92e13T2 1 + 4.14e6iT - 1.92e13T^{2}
83 1+(2.77e6+2.77e6i)T+2.71e13iT2 1 + (2.77e6 + 2.77e6i)T + 2.71e13iT^{2}
89 1−1.02e7T+4.42e13T2 1 - 1.02e7T + 4.42e13T^{2}
97 1+(−2.22e6−2.22e6i)T+8.07e13iT2 1 + (-2.22e6 - 2.22e6i)T + 8.07e13iT^{2}
show more
show less
   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

−11.75293645922753890698686786615, −10.51295925303438223309696427330, −9.994107175821207940169412941934, −8.721719466378078161458722557384, −6.96650396153958170518761776744, −5.95002876317506557430244380683, −4.74634901432683935268521569946, −3.94664815632819468127401970909, −2.27089685178641427114501421654, −0.62716331060160982821928321731, 0.829473304388402697881820578639, 2.57167691343378046309250916080, 4.26964439191734611822218463431, 5.48064755253146867873945573981, 6.29859875087767801105748409451, 7.39024921722491480047338229621, 8.320559992285198471004966510449, 9.852740644062403149430343305546, 11.15682463573785630025036262667, 11.85606223255835777977656076815

Graph of the ZZ-function along the critical line