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Properties

Label 2-3240-360.29-c0-0-6
Degree 22
Conductor 32403240
Sign 0.984+0.173i0.984 + 0.173i
Analytic cond. 1.616971.61697
Root an. cond. 1.271601.27160
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·8-s + 0.999·10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s − 17-s + (−0.499 + 0.866i)20-s + (0.499 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s − 0.999·26-s + (0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·8-s + 0.999·10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s − 17-s + (−0.499 + 0.866i)20-s + (0.499 + 0.866i)22-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s − 0.999·26-s + (0.5 − 0.866i)29-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.984+0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s) L(s)=((0.984+0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.984+0.173i0.984 + 0.173i
Analytic conductor: 1.616971.61697
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3240(269,⋅)\chi_{3240} (269, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :0), 0.984+0.173i)(2,\ 3240,\ (\ :0),\ 0.984 + 0.173i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.83408456760.8340845676
L(12)L(\frac12) ≈\approx 0.83408456760.8340845676
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1 1
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
13 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1+T+T2 1 + T + T^{2}
19 1−T2 1 - T^{2}
23 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
29 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
31 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
37 1−2T+T2 1 - 2T + T^{2}
41 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
43 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
47 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1−T2 1 - T^{2}
59 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
61 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
67 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)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

−8.847256767773727751890504112691, −8.147067349588260427593720587206, −7.43696523056642020355814386797, −6.50345091691485241019866969658, −6.03697320626069672989523305737, −4.99029092266473846364041982977, −4.38705127956497898877064514643, −3.53924594738823898427324411818, −1.85521153533964803228598516297, −0.74772087896610239237699025109, 1.11556314710997276798060699167, 2.53906348411582632998367519298, 2.95879858435625949332144194527, 4.24468236848494426573252661155, 4.46155957936761238878387134206, 6.00436858070578905170730733991, 6.78727251764810709667060603029, 7.55553668859962581290153335529, 8.125122896397652423286717724036, 8.979740708451834896787522880861

Graph of the ZZ-function along the critical line