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Properties

Label 2-30e2-300.23-c0-0-0
Degree 22
Conductor 900900
Sign 0.720−0.693i0.720 - 0.693i
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  + (0.987 + 0.156i)2-s + (0.951 + 0.309i)4-s + (−0.453 + 0.891i)5-s + (0.891 + 0.453i)8-s + (−0.587 + 0.809i)10-s + (−0.142 − 0.896i)13-s + (0.809 + 0.587i)16-s + (−0.533 + 1.04i)17-s + (−0.707 + 0.707i)20-s + (−0.587 − 0.809i)25-s − 0.907i·26-s + (0.610 − 1.87i)29-s + (0.707 + 0.707i)32-s + (−0.690 + 0.951i)34-s + (−0.309 + 0.0489i)37-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (0.951 + 0.309i)4-s + (−0.453 + 0.891i)5-s + (0.891 + 0.453i)8-s + (−0.587 + 0.809i)10-s + (−0.142 − 0.896i)13-s + (0.809 + 0.587i)16-s + (−0.533 + 1.04i)17-s + (−0.707 + 0.707i)20-s + (−0.587 − 0.809i)25-s − 0.907i·26-s + (0.610 − 1.87i)29-s + (0.707 + 0.707i)32-s + (−0.690 + 0.951i)34-s + (−0.309 + 0.0489i)37-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.720−0.693i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s) L(s)=((0.720−0.693i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.720−0.693i0.720 - 0.693i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(323,⋅)\chi_{900} (323, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :0), 0.720−0.693i)(2,\ 900,\ (\ :0),\ 0.720 - 0.693i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6785759461.678575946
L(12)L(\frac12) ≈\approx 1.6785759461.678575946
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+(−0.987−0.156i)T 1 + (-0.987 - 0.156i)T
3 1 1
5 1+(0.453−0.891i)T 1 + (0.453 - 0.891i)T
good7 1+iT2 1 + iT^{2}
11 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
13 1+(0.142+0.896i)T+(−0.951+0.309i)T2 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2}
17 1+(0.533−1.04i)T+(−0.587−0.809i)T2 1 + (0.533 - 1.04i)T + (-0.587 - 0.809i)T^{2}
19 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
23 1+(0.951+0.309i)T2 1 + (0.951 + 0.309i)T^{2}
29 1+(−0.610+1.87i)T+(−0.809−0.587i)T2 1 + (-0.610 + 1.87i)T + (-0.809 - 0.587i)T^{2}
31 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
37 1+(0.309−0.0489i)T+(0.951−0.309i)T2 1 + (0.309 - 0.0489i)T + (0.951 - 0.309i)T^{2}
41 1+(1.04−1.44i)T+(−0.309−0.951i)T2 1 + (1.04 - 1.44i)T + (-0.309 - 0.951i)T^{2}
43 1−iT2 1 - iT^{2}
47 1+(−0.587+0.809i)T2 1 + (-0.587 + 0.809i)T^{2}
53 1+(0.863+1.69i)T+(−0.587+0.809i)T2 1 + (0.863 + 1.69i)T + (-0.587 + 0.809i)T^{2}
59 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
61 1+(−1.53+1.11i)T+(0.309−0.951i)T2 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2}
67 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
71 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
73 1+(1.76+0.278i)T+(0.951+0.309i)T2 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2}
79 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
83 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
89 1+(−0.253+0.183i)T+(0.309−0.951i)T2 1 + (-0.253 + 0.183i)T + (0.309 - 0.951i)T^{2}
97 1+(−0.896−1.76i)T+(−0.587+0.809i)T2 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)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

−10.50881760726300515051965708608, −9.977947496226990216364125933939, −8.268584491802090880412261008462, −7.87164508951141728687826455595, −6.71119540166231237884624369981, −6.23184802219006896851038236338, −5.10635235600885901129405361651, −4.04742927709698840884975793573, −3.22932875204371312786881908952, −2.17929036591852459450067450580, 1.52614360141319530145078660978, 2.91594584388946809654208969471, 4.09832610009877531153763933818, 4.80071572186294648993526789698, 5.57078962052024463958003650785, 6.82342914285369257813225379827, 7.37004033163099433186580630751, 8.629202309632283597076948980986, 9.305020314635783967264084711832, 10.43217491404989629776185589054

Graph of the ZZ-function along the critical line