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Properties

Label 2-3216-201.158-c0-0-0
Degree 22
Conductor 32163216
Sign −0.463+0.886i-0.463 + 0.886i
Analytic cond. 1.604991.60499
Root an. cond. 1.266881.26688
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  + (−0.841 + 0.540i)3-s + (0.544 − 1.19i)7-s + (0.415 − 0.909i)9-s + (−0.797 − 0.234i)13-s + (−0.698 − 1.53i)19-s + (0.186 + 1.29i)21-s + (−0.959 − 0.281i)25-s + (0.142 + 0.989i)27-s + (−0.273 + 0.0801i)31-s − 1.91·37-s + (0.797 − 0.234i)39-s + (−1.25 + 1.45i)43-s + (−0.468 − 0.540i)49-s + (1.41 + 0.909i)57-s + (−0.239 − 1.66i)61-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)3-s + (0.544 − 1.19i)7-s + (0.415 − 0.909i)9-s + (−0.797 − 0.234i)13-s + (−0.698 − 1.53i)19-s + (0.186 + 1.29i)21-s + (−0.959 − 0.281i)25-s + (0.142 + 0.989i)27-s + (−0.273 + 0.0801i)31-s − 1.91·37-s + (0.797 − 0.234i)39-s + (−1.25 + 1.45i)43-s + (−0.468 − 0.540i)49-s + (1.41 + 0.909i)57-s + (−0.239 − 1.66i)61-s + ⋯

Functional equation

Λ(s)=(3216s/2 ΓC(s) L(s)=((−0.463+0.886i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3216s/2 ΓC(s) L(s)=((−0.463+0.886i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32163216    =    24⋅3⋅672^{4} \cdot 3 \cdot 67
Sign: −0.463+0.886i-0.463 + 0.886i
Analytic conductor: 1.604991.60499
Root analytic conductor: 1.266881.26688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3216(2369,⋅)\chi_{3216} (2369, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3216, ( :0), −0.463+0.886i)(2,\ 3216,\ (\ :0),\ -0.463 + 0.886i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.52897965460.5289796546
L(12)L(\frac12) ≈\approx 0.52897965460.5289796546
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+(0.841−0.540i)T 1 + (0.841 - 0.540i)T
67 1+(−0.959−0.281i)T 1 + (-0.959 - 0.281i)T
good5 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
7 1+(−0.544+1.19i)T+(−0.654−0.755i)T2 1 + (-0.544 + 1.19i)T + (-0.654 - 0.755i)T^{2}
11 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
13 1+(0.797+0.234i)T+(0.841+0.540i)T2 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2}
17 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
19 1+(0.698+1.53i)T+(−0.654+0.755i)T2 1 + (0.698 + 1.53i)T + (-0.654 + 0.755i)T^{2}
23 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.273−0.0801i)T+(0.841−0.540i)T2 1 + (0.273 - 0.0801i)T + (0.841 - 0.540i)T^{2}
37 1+1.91T+T2 1 + 1.91T + T^{2}
41 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
43 1+(1.25−1.45i)T+(−0.142−0.989i)T2 1 + (1.25 - 1.45i)T + (-0.142 - 0.989i)T^{2}
47 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
53 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)T^{2}
59 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
61 1+(0.239+1.66i)T+(−0.959+0.281i)T2 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2}
71 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)T^{2}
73 1+(−0.0405−0.281i)T+(−0.959+0.281i)T2 1 + (-0.0405 - 0.281i)T + (-0.959 + 0.281i)T^{2}
79 1+(1.84+0.540i)T+(0.841+0.540i)T2 1 + (1.84 + 0.540i)T + (0.841 + 0.540i)T^{2}
83 1+(0.959+0.281i)T2 1 + (0.959 + 0.281i)T^{2}
89 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
97 1−1.68T+T2 1 - 1.68T + 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

−8.637322616264086959614104007244, −7.72702612716764118367969778477, −6.98981155769894544185729458737, −6.46699177001312184749168055445, −5.32954934096106080269834046967, −4.74863134224319635961267709234, −4.14234991907198829521347885934, −3.16813709155864463561006115575, −1.74648742397330497905908348314, −0.33948603374216275500986429955, 1.74032095961211331070223656552, 2.18992389403935612858383460358, 3.63760592673497916987219729381, 4.71625390051147910156741035805, 5.47997952967927170134077541226, 5.87908681036312086617584730975, 6.83424994413530122483504164231, 7.53396581713877216343582175223, 8.329212871271315056946017739926, 8.888351268132571993043996057513

Graph of the ZZ-function along the critical line