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Properties

Label 2-30e2-300.47-c0-0-0
Degree 22
Conductor 900900
Sign −0.762−0.647i-0.762 - 0.647i
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  + (0.453 + 0.891i)2-s + (−0.587 + 0.809i)4-s + (−0.156 + 0.987i)5-s + (−0.987 − 0.156i)8-s + (−0.951 + 0.309i)10-s + (0.278 + 0.142i)13-s + (−0.309 − 0.951i)16-s + (−0.297 + 1.87i)17-s + (−0.707 − 0.707i)20-s + (−0.951 − 0.309i)25-s + 0.312i·26-s + (−0.734 − 0.533i)29-s + (0.707 − 0.707i)32-s + (−1.80 + 0.587i)34-s + (0.809 − 1.58i)37-s + ⋯
L(s)  = 1  + (0.453 + 0.891i)2-s + (−0.587 + 0.809i)4-s + (−0.156 + 0.987i)5-s + (−0.987 − 0.156i)8-s + (−0.951 + 0.309i)10-s + (0.278 + 0.142i)13-s + (−0.309 − 0.951i)16-s + (−0.297 + 1.87i)17-s + (−0.707 − 0.707i)20-s + (−0.951 − 0.309i)25-s + 0.312i·26-s + (−0.734 − 0.533i)29-s + (0.707 − 0.707i)32-s + (−1.80 + 0.587i)34-s + (0.809 − 1.58i)37-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((−0.762−0.647i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s) L(s)=((−0.762−0.647i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: −0.762−0.647i-0.762 - 0.647i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(647,⋅)\chi_{900} (647, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :0), −0.762−0.647i)(2,\ 900,\ (\ :0),\ -0.762 - 0.647i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0644687941.064468794
L(12)L(\frac12) ≈\approx 1.0644687941.064468794
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+(−0.453−0.891i)T 1 + (-0.453 - 0.891i)T
3 1 1
5 1+(0.156−0.987i)T 1 + (0.156 - 0.987i)T
good7 1−iT2 1 - iT^{2}
11 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
13 1+(−0.278−0.142i)T+(0.587+0.809i)T2 1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2}
17 1+(0.297−1.87i)T+(−0.951−0.309i)T2 1 + (0.297 - 1.87i)T + (-0.951 - 0.309i)T^{2}
19 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
23 1+(−0.587+0.809i)T2 1 + (-0.587 + 0.809i)T^{2}
29 1+(0.734+0.533i)T+(0.309+0.951i)T2 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2}
31 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
37 1+(−0.809+1.58i)T+(−0.587−0.809i)T2 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2}
41 1+(−1.87+0.610i)T+(0.809−0.587i)T2 1 + (-1.87 + 0.610i)T + (0.809 - 0.587i)T^{2}
43 1+iT2 1 + iT^{2}
47 1+(−0.951+0.309i)T2 1 + (-0.951 + 0.309i)T^{2}
53 1+(−0.183−1.16i)T+(−0.951+0.309i)T2 1 + (-0.183 - 1.16i)T + (-0.951 + 0.309i)T^{2}
59 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
61 1+(−0.363+1.11i)T+(−0.809−0.587i)T2 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2}
67 1+(−0.951−0.309i)T2 1 + (-0.951 - 0.309i)T^{2}
71 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
73 1+(−0.896−1.76i)T+(−0.587+0.809i)T2 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2}
79 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
83 1+(−0.951−0.309i)T2 1 + (-0.951 - 0.309i)T^{2}
89 1+(−0.550+1.69i)T+(−0.809−0.587i)T2 1 + (-0.550 + 1.69i)T + (-0.809 - 0.587i)T^{2}
97 1+(−0.142−0.896i)T+(−0.951+0.309i)T2 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)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

−10.77715620642285723043289191784, −9.677549995403535994008589934281, −8.775924941422143080629292112353, −7.84908334251719834592495512030, −7.25636241112901354279208129536, −6.15938622320798448766860038813, −5.84006726057176150472361588747, −4.25822353329832922846133460837, −3.69847492338357884506650748840, −2.37899446072060370362761870870, 0.968637266781333851888623831607, 2.42427285718060389579826026736, 3.59585463147861824728990141378, 4.67867346131308493841329757364, 5.21980349838940910140709753271, 6.29509474576031046504658309863, 7.55470586932254124357804731271, 8.602916110938932053267267472698, 9.336045545715528348276387935845, 9.897966698847325220363718328858

Graph of the ZZ-function along the critical line