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Properties

Label 2-30e2-180.23-c0-0-1
Degree 22
Conductor 900900
Sign 0.979−0.203i0.979 - 0.203i
Analytic cond. 0.4491580.449158
Root an. cond. 0.6701920.670192
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.965 + 0.258i)3-s + (0.866 + 0.499i)4-s − 6-s + (0.448 − 1.67i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (−0.965 − 0.258i)12-s + (0.866 − 1.50i)14-s + (0.500 + 0.866i)16-s + (0.965 − 0.258i)18-s + 1.73i·21-s + (−0.965 + 0.258i)23-s + (−0.866 − 0.5i)24-s + (−0.707 + 0.707i)27-s + (1.22 − 1.22i)28-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.965 + 0.258i)3-s + (0.866 + 0.499i)4-s − 6-s + (0.448 − 1.67i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (−0.965 − 0.258i)12-s + (0.866 − 1.50i)14-s + (0.500 + 0.866i)16-s + (0.965 − 0.258i)18-s + 1.73i·21-s + (−0.965 + 0.258i)23-s + (−0.866 − 0.5i)24-s + (−0.707 + 0.707i)27-s + (1.22 − 1.22i)28-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.979−0.203i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s) L(s)=((0.979−0.203i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.979−0.203i0.979 - 0.203i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(743,⋅)\chi_{900} (743, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :0), 0.979−0.203i)(2,\ 900,\ (\ :0),\ 0.979 - 0.203i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4154112831.415411283
L(12)L(\frac12) ≈\approx 1.4154112831.415411283
L(1)L(1) 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+(−0.965−0.258i)T 1 + (-0.965 - 0.258i)T
3 1+(0.965−0.258i)T 1 + (0.965 - 0.258i)T
5 1 1
good7 1+(−0.448+1.67i)T+(−0.866−0.5i)T2 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2}
11 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
13 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
17 1+iT2 1 + iT^{2}
19 1+T2 1 + T^{2}
23 1+(0.965−0.258i)T+(0.866−0.5i)T2 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2}
29 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−iT2 1 - iT^{2}
41 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
43 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
47 1+(−0.965−0.258i)T+(0.866+0.5i)T2 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2}
53 1−iT2 1 - iT^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(1.67−0.448i)T+(0.866−0.5i)T2 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2}
71 1+T2 1 + T^{2}
73 1+iT2 1 + iT^{2}
79 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
83 1+(0.258−0.965i)T+(−0.866−0.5i)T2 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2}
89 1+1.73T+T2 1 + 1.73T + T^{2}
97 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)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

−10.61024084148005872853728023029, −9.954811926163750646723557801932, −8.406894277197425139893893103049, −7.32072047898068679310585424246, −6.92197437504010791210308431352, −5.90615777641067929495360424540, −4.93889115701135289223646000387, −4.26439175718954740147527536408, −3.44535657682999888105332903457, −1.48653732501967385173032576407, 1.75272355592103926429228939309, 2.71510360452837235332501017325, 4.28726665817755760811856770182, 5.08773990719179494790862416046, 5.91054678387368203948835812751, 6.35007321993085162908791825440, 7.56171563207534452048257380197, 8.518442638723591320252741323745, 9.771875878425240809348035258557, 10.49820389654922155896267784016

Graph of the ZZ-function along the critical line