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Properties

Label 2-150-25.21-c1-0-2
Degree 22
Conductor 150150
Sign 0.927+0.373i0.927 + 0.373i
Analytic cond. 1.197751.19775
Root an. cond. 1.094421.09442
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.00655 − 2.23i)5-s + (0.809 − 0.587i)6-s + 2.63·7-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (2.12 + 0.697i)10-s + (1.93 − 5.96i)11-s + (0.309 + 0.951i)12-s + (0.697 + 2.14i)13-s + (−0.815 + 2.51i)14-s + (−1.31 + 1.80i)15-s + (0.309 + 0.951i)16-s + (−1.31 + 0.955i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.00293 − 0.999i)5-s + (0.330 − 0.239i)6-s + 0.997·7-s + (0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + (0.671 + 0.220i)10-s + (0.584 − 1.79i)11-s + (0.0892 + 0.274i)12-s + (0.193 + 0.595i)13-s + (−0.217 + 0.670i)14-s + (−0.340 + 0.466i)15-s + (0.0772 + 0.237i)16-s + (−0.319 + 0.231i)17-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((0.927+0.373i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+1/2) L(s)=((0.927+0.373i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: 0.927+0.373i0.927 + 0.373i
Analytic conductor: 1.197751.19775
Root analytic conductor: 1.094421.09442
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ150(121,⋅)\chi_{150} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :1/2), 0.927+0.373i)(2,\ 150,\ (\ :1/2),\ 0.927 + 0.373i)

Particular Values

L(1)L(1) ≈\approx 0.899413−0.174307i0.899413 - 0.174307i
L(12)L(\frac12) ≈\approx 0.899413−0.174307i0.899413 - 0.174307i
L(32)L(\frac{3}{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+(0.309−0.951i)T 1 + (0.309 - 0.951i)T
3 1+(0.809+0.587i)T 1 + (0.809 + 0.587i)T
5 1+(−0.00655+2.23i)T 1 + (-0.00655 + 2.23i)T
good7 1−2.63T+7T2 1 - 2.63T + 7T^{2}
11 1+(−1.93+5.96i)T+(−8.89−6.46i)T2 1 + (-1.93 + 5.96i)T + (-8.89 - 6.46i)T^{2}
13 1+(−0.697−2.14i)T+(−10.5+7.64i)T2 1 + (-0.697 - 2.14i)T + (-10.5 + 7.64i)T^{2}
17 1+(1.31−0.955i)T+(5.25−16.1i)T2 1 + (1.31 - 0.955i)T + (5.25 - 16.1i)T^{2}
19 1+(−1+0.726i)T+(5.87−18.0i)T2 1 + (-1 + 0.726i)T + (5.87 - 18.0i)T^{2}
23 1+(1.38−4.25i)T+(−18.6−13.5i)T2 1 + (1.38 - 4.25i)T + (-18.6 - 13.5i)T^{2}
29 1+(5.55+4.03i)T+(8.96+27.5i)T2 1 + (5.55 + 4.03i)T + (8.96 + 27.5i)T^{2}
31 1+(−4.75+3.45i)T+(9.57−29.4i)T2 1 + (-4.75 + 3.45i)T + (9.57 - 29.4i)T^{2}
37 1+(−2.49−7.68i)T+(−29.9+21.7i)T2 1 + (-2.49 - 7.68i)T + (-29.9 + 21.7i)T^{2}
41 1+(−3.55−10.9i)T+(−33.1+24.0i)T2 1 + (-3.55 - 10.9i)T + (-33.1 + 24.0i)T^{2}
43 1+1.97T+43T2 1 + 1.97T + 43T^{2}
47 1+(5.25+3.81i)T+(14.5+44.6i)T2 1 + (5.25 + 3.81i)T + (14.5 + 44.6i)T^{2}
53 1+(−5.66−4.11i)T+(16.3+50.4i)T2 1 + (-5.66 - 4.11i)T + (16.3 + 50.4i)T^{2}
59 1+(−2.79−8.59i)T+(−47.7+34.6i)T2 1 + (-2.79 - 8.59i)T + (-47.7 + 34.6i)T^{2}
61 1+(0.933−2.87i)T+(−49.3−35.8i)T2 1 + (0.933 - 2.87i)T + (-49.3 - 35.8i)T^{2}
67 1+(−8.12+5.90i)T+(20.7−63.7i)T2 1 + (-8.12 + 5.90i)T + (20.7 - 63.7i)T^{2}
71 1+(−7.90−5.74i)T+(21.9+67.5i)T2 1 + (-7.90 - 5.74i)T + (21.9 + 67.5i)T^{2}
73 1+(3.70−11.4i)T+(−59.0−42.9i)T2 1 + (3.70 - 11.4i)T + (-59.0 - 42.9i)T^{2}
79 1+(3.03+2.20i)T+(24.4+75.1i)T2 1 + (3.03 + 2.20i)T + (24.4 + 75.1i)T^{2}
83 1+(−6.77+4.92i)T+(25.6−78.9i)T2 1 + (-6.77 + 4.92i)T + (25.6 - 78.9i)T^{2}
89 1+(0.120−0.370i)T+(−72.0−52.3i)T2 1 + (0.120 - 0.370i)T + (-72.0 - 52.3i)T^{2}
97 1+(1.21+0.882i)T+(29.9+92.2i)T2 1 + (1.21 + 0.882i)T + (29.9 + 92.2i)T^{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

−13.31006606409504743272694517554, −11.62794365915299264393793094882, −11.35701731242394527239408762338, −9.615563685170127233446338075854, −8.522335688744759306742220518208, −7.921397933557687455571201214919, −6.33458672652040128067310709995, −5.45710638119489757845118112524, −4.22323358205086127576466658121, −1.22786740220948346480124704589, 2.08959845916282485844296962097, 3.91396595650554236474837334782, 5.09864949159950742781207146426, 6.77117178982815128573226386239, 7.81371276936030481610151480357, 9.299231046524535742511740319763, 10.28911120820932503095894500756, 10.97611231495990210945161462850, 11.87988356423732047425947957808, 12.73325313603969579184911857548

Graph of the ZZ-function along the critical line