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Properties

Label 2-150-25.16-c1-0-3
Degree 22
Conductor 150150
Sign 0.992+0.125i0.992 + 0.125i
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  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.690 − 2.12i)5-s + (0.309 − 0.951i)6-s + 2·7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (1.80 − 1.31i)10-s + (0.618 + 0.449i)11-s + (0.809 − 0.587i)12-s + (−1.5 + 1.08i)13-s + (1.61 + 1.17i)14-s − 2.23·15-s + (−0.809 + 0.587i)16-s + (0.354 − 1.08i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (0.309 − 0.951i)5-s + (0.126 − 0.388i)6-s + 0.755·7-s + (−0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.572 − 0.415i)10-s + (0.186 + 0.135i)11-s + (0.233 − 0.169i)12-s + (−0.416 + 0.302i)13-s + (0.432 + 0.314i)14-s − 0.577·15-s + (−0.202 + 0.146i)16-s + (0.0858 − 0.264i)17-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((0.992+0.125i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+1/2) L(s)=((0.992+0.125i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: 0.992+0.125i0.992 + 0.125i
Analytic conductor: 1.197751.19775
Root analytic conductor: 1.094421.09442
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ150(91,⋅)\chi_{150} (91, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :1/2), 0.992+0.125i)(2,\ 150,\ (\ :1/2),\ 0.992 + 0.125i)

Particular Values

L(1)L(1) ≈\approx 1.49453−0.0940282i1.49453 - 0.0940282i
L(12)L(\frac12) ≈\approx 1.49453−0.0940282i1.49453 - 0.0940282i
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+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
3 1+(0.309+0.951i)T 1 + (0.309 + 0.951i)T
5 1+(−0.690+2.12i)T 1 + (-0.690 + 2.12i)T
good7 1−2T+7T2 1 - 2T + 7T^{2}
11 1+(−0.618−0.449i)T+(3.39+10.4i)T2 1 + (-0.618 - 0.449i)T + (3.39 + 10.4i)T^{2}
13 1+(1.5−1.08i)T+(4.01−12.3i)T2 1 + (1.5 - 1.08i)T + (4.01 - 12.3i)T^{2}
17 1+(−0.354+1.08i)T+(−13.7−9.99i)T2 1 + (-0.354 + 1.08i)T + (-13.7 - 9.99i)T^{2}
19 1+(2.23−6.88i)T+(−15.3−11.1i)T2 1 + (2.23 - 6.88i)T + (-15.3 - 11.1i)T^{2}
23 1+(4.85+3.52i)T+(7.10+21.8i)T2 1 + (4.85 + 3.52i)T + (7.10 + 21.8i)T^{2}
29 1+(1.11+3.44i)T+(−23.4+17.0i)T2 1 + (1.11 + 3.44i)T + (-23.4 + 17.0i)T^{2}
31 1+(3−9.23i)T+(−25.0−18.2i)T2 1 + (3 - 9.23i)T + (-25.0 - 18.2i)T^{2}
37 1+(−7.16+5.20i)T+(11.4−35.1i)T2 1 + (-7.16 + 5.20i)T + (11.4 - 35.1i)T^{2}
41 1+(4.11−2.99i)T+(12.6−38.9i)T2 1 + (4.11 - 2.99i)T + (12.6 - 38.9i)T^{2}
43 1−3.23T+43T2 1 - 3.23T + 43T^{2}
47 1+(−2.85−8.78i)T+(−38.0+27.6i)T2 1 + (-2.85 - 8.78i)T + (-38.0 + 27.6i)T^{2}
53 1+(3.57+10.9i)T+(−42.8+31.1i)T2 1 + (3.57 + 10.9i)T + (-42.8 + 31.1i)T^{2}
59 1+(−7.23+5.25i)T+(18.2−56.1i)T2 1 + (-7.23 + 5.25i)T + (18.2 - 56.1i)T^{2}
61 1+(−1.73−1.26i)T+(18.8+58.0i)T2 1 + (-1.73 - 1.26i)T + (18.8 + 58.0i)T^{2}
67 1+(−1.14+3.52i)T+(−54.2−39.3i)T2 1 + (-1.14 + 3.52i)T + (-54.2 - 39.3i)T^{2}
71 1+(−2.52−7.77i)T+(−57.4+41.7i)T2 1 + (-2.52 - 7.77i)T + (-57.4 + 41.7i)T^{2}
73 1+(−7.97−5.79i)T+(22.5+69.4i)T2 1 + (-7.97 - 5.79i)T + (22.5 + 69.4i)T^{2}
79 1+(−63.9+46.4i)T2 1 + (-63.9 + 46.4i)T^{2}
83 1+(−1.85+5.70i)T+(−67.1−48.7i)T2 1 + (-1.85 + 5.70i)T + (-67.1 - 48.7i)T^{2}
89 1+(2.92+2.12i)T+(27.5+84.6i)T2 1 + (2.92 + 2.12i)T + (27.5 + 84.6i)T^{2}
97 1+(2.20+6.79i)T+(−78.4+57.0i)T2 1 + (2.20 + 6.79i)T + (-78.4 + 57.0i)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

−12.81728510557522293120648429568, −12.34290600349028552178431530951, −11.36033808836525419913030935056, −9.884079659220647862020962358029, −8.499241886291921564136946711727, −7.76726723566014285915545147718, −6.36060986634045218351650888427, −5.31749180948017034830304535510, −4.21501948897060548742146284126, −1.89414489831281707489994724004, 2.39424812705518774258246690486, 3.88651645387973616330895132183, 5.18463910985769388165492696502, 6.31836009160644899111225545213, 7.64982664406823497799484814159, 9.243439402267130846148126510136, 10.26938872061852166842472006831, 11.12197011188085210931820762365, 11.73524015727050511716928480188, 13.16615254079391933567342796186

Graph of the ZZ-function along the critical line