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Properties

Label 2-30e2-300.203-c0-0-1
Degree 22
Conductor 900900
Sign 0.762+0.647i0.762 + 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  + (0.891 − 0.453i)2-s + (0.587 − 0.809i)4-s + (0.987 + 0.156i)5-s + (0.156 − 0.987i)8-s + (0.951 − 0.309i)10-s + (−0.896 + 1.76i)13-s + (−0.309 − 0.951i)16-s + (−1.87 − 0.297i)17-s + (0.707 − 0.707i)20-s + (0.951 + 0.309i)25-s + 1.97i·26-s + (−1.44 − 1.04i)29-s + (−0.707 − 0.707i)32-s + (−1.80 + 0.587i)34-s + (0.809 + 0.412i)37-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)2-s + (0.587 − 0.809i)4-s + (0.987 + 0.156i)5-s + (0.156 − 0.987i)8-s + (0.951 − 0.309i)10-s + (−0.896 + 1.76i)13-s + (−0.309 − 0.951i)16-s + (−1.87 − 0.297i)17-s + (0.707 − 0.707i)20-s + (0.951 + 0.309i)25-s + 1.97i·26-s + (−1.44 − 1.04i)29-s + (−0.707 − 0.707i)32-s + (−1.80 + 0.587i)34-s + (0.809 + 0.412i)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+0.647i0.762 + 0.647i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(503,⋅)\chi_{900} (503, \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 1.7834839011.783483901
L(12)L(\frac12) ≈\approx 1.7834839011.783483901
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+(−0.891+0.453i)T 1 + (-0.891 + 0.453i)T
3 1 1
5 1+(−0.987−0.156i)T 1 + (-0.987 - 0.156i)T
good7 1+iT2 1 + iT^{2}
11 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
13 1+(0.896−1.76i)T+(−0.587−0.809i)T2 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2}
17 1+(1.87+0.297i)T+(0.951+0.309i)T2 1 + (1.87 + 0.297i)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+(1.44+1.04i)T+(0.309+0.951i)T2 1 + (1.44 + 1.04i)T + (0.309 + 0.951i)T^{2}
31 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
37 1+(−0.809−0.412i)T+(0.587+0.809i)T2 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2}
41 1+(−0.297+0.0966i)T+(0.809−0.587i)T2 1 + (-0.297 + 0.0966i)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+(−1.16+0.183i)T+(0.951−0.309i)T2 1 + (-1.16 + 0.183i)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+(0.278−0.142i)T+(0.587−0.809i)T2 1 + (0.278 - 0.142i)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+(−0.280+0.863i)T+(−0.809−0.587i)T2 1 + (-0.280 + 0.863i)T + (-0.809 - 0.587i)T^{2}
97 1+(1.76−0.278i)T+(0.951−0.309i)T2 1 + (1.76 - 0.278i)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

−10.26200786481254251915715518334, −9.495164798497544813885698392455, −8.961132016459778387106406467879, −7.22408626738866596027990801597, −6.65431196213229251477929329147, −5.82409855472728154804306542907, −4.76275111215434283344644030924, −4.08292346278167262587104018299, −2.48451853843052787941143155655, −1.94411884950686831028261501544, 2.10943664465414073280822298894, 3.01163886995903842155519242580, 4.36469933681150109980861661041, 5.27942561312977249183062576581, 5.89526077755966659292296768195, 6.84713562938236125333663758988, 7.68104740642207546812769486796, 8.655757768446618748027378154981, 9.505827637465213051821098236912, 10.62468654645593219656399147934

Graph of the ZZ-function along the critical line