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Properties

Label 2-60e2-300.167-c0-0-0
Degree 22
Conductor 36003600
Sign −0.0755−0.997i-0.0755 - 0.997i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
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  + (0.156 + 0.987i)5-s + (0.896 + 1.76i)13-s + (−0.610 + 0.0966i)17-s + (−0.951 + 0.309i)25-s + (−0.734 + 0.533i)29-s + (0.809 − 0.412i)37-s + (−1.87 − 0.610i)41-s + i·49-s + (1.59 + 0.253i)53-s + (−0.363 − 1.11i)61-s + (−1.59 + 1.16i)65-s + (0.278 + 0.142i)73-s + (−0.190 − 0.587i)85-s + (0.550 + 1.69i)89-s + (1.76 + 0.278i)97-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)5-s + (0.896 + 1.76i)13-s + (−0.610 + 0.0966i)17-s + (−0.951 + 0.309i)25-s + (−0.734 + 0.533i)29-s + (0.809 − 0.412i)37-s + (−1.87 − 0.610i)41-s + i·49-s + (1.59 + 0.253i)53-s + (−0.363 − 1.11i)61-s + (−1.59 + 1.16i)65-s + (0.278 + 0.142i)73-s + (−0.190 − 0.587i)85-s + (0.550 + 1.69i)89-s + (1.76 + 0.278i)97-s + ⋯

Functional equation

Λ(s)=(3600s/2 ΓC(s) L(s)=((−0.0755−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0755 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3600s/2 ΓC(s) L(s)=((−0.0755−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0755 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36003600    =    24⋅32⋅522^{4} \cdot 3^{2} \cdot 5^{2}
Sign: −0.0755−0.997i-0.0755 - 0.997i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3600(3167,⋅)\chi_{3600} (3167, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3600, ( :0), −0.0755−0.997i)(2,\ 3600,\ (\ :0),\ -0.0755 - 0.997i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1967241701.196724170
L(12)L(\frac12) ≈\approx 1.1967241701.196724170
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 1
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+(−0.896−1.76i)T+(−0.587+0.809i)T2 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2}
17 1+(0.610−0.0966i)T+(0.951−0.309i)T2 1 + (0.610 - 0.0966i)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+0.412i)T+(0.587−0.809i)T2 1 + (-0.809 + 0.412i)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+(−1.59−0.253i)T+(0.951+0.309i)T2 1 + (-1.59 - 0.253i)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+(−0.550−1.69i)T+(−0.809+0.587i)T2 1 + (-0.550 - 1.69i)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

−9.079958101656393563536377603060, −8.192251330977358747064160232755, −7.26744890328745977725775672200, −6.66529779717285786799880094166, −6.19224062830832542149036722516, −5.19732219769330681461946721530, −4.11580368540707788380055640994, −3.58551380680103488750504367361, −2.41947158930724095630113378281, −1.64145523951003654045980133972, 0.69900473873621712118897565285, 1.84821757138007885464556408404, 3.03754286727657892250559330378, 3.91106776175062514125337173662, 4.81310247356433936247705298518, 5.55976092386691760361522151260, 6.08978635254481733153249457871, 7.12179458604650262776765993068, 8.062231833181424253803742489624, 8.437158815216965625003277849061

Graph of the ZZ-function along the critical line