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Properties

Label 2-1368-1368.283-c0-0-1
Degree 22
Conductor 13681368
Sign −0.776+0.630i-0.776 + 0.630i
Analytic cond. 0.6827200.682720
Root an. cond. 0.8262690.826269
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  + (−0.939 + 0.342i)2-s + (−0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + 6-s + (−0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s − 1.87·11-s + (−0.939 + 0.342i)12-s + (0.173 − 0.984i)16-s + (0.347 − 1.96i)17-s + (−0.939 − 0.342i)18-s + (0.173 + 0.984i)19-s + (1.76 − 0.642i)22-s + (0.766 − 0.642i)24-s + (−0.939 − 0.342i)25-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + 6-s + (−0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s − 1.87·11-s + (−0.939 + 0.342i)12-s + (0.173 − 0.984i)16-s + (0.347 − 1.96i)17-s + (−0.939 − 0.342i)18-s + (0.173 + 0.984i)19-s + (1.76 − 0.642i)22-s + (0.766 − 0.642i)24-s + (−0.939 − 0.342i)25-s + ⋯

Functional equation

Λ(s)=(1368s/2 ΓC(s) L(s)=((−0.776+0.630i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1368s/2 ΓC(s) L(s)=((−0.776+0.630i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13681368    =    23⋅32⋅192^{3} \cdot 3^{2} \cdot 19
Sign: −0.776+0.630i-0.776 + 0.630i
Analytic conductor: 0.6827200.682720
Root analytic conductor: 0.8262690.826269
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1368(283,⋅)\chi_{1368} (283, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1368, ( :0), −0.776+0.630i)(2,\ 1368,\ (\ :0),\ -0.776 + 0.630i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.16224645140.1622464514
L(12)L(\frac12) ≈\approx 0.16224645140.1622464514
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+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
3 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
19 1+(−0.173−0.984i)T 1 + (-0.173 - 0.984i)T
good5 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
7 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+1.87T+T2 1 + 1.87T + T^{2}
13 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
17 1+(−0.347+1.96i)T+(−0.939−0.342i)T2 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2}
23 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
29 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
31 1−T2 1 - T^{2}
37 1−T2 1 - T^{2}
41 1+(1.43−0.524i)T+(0.766−0.642i)T2 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2}
43 1+(0.766+0.642i)T+(0.173+0.984i)T2 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2}
47 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
53 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
59 1+(−0.266−0.223i)T+(0.173+0.984i)T2 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2}
61 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
67 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2}
71 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
73 1+(1.43+1.20i)T+(0.173+0.984i)T2 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2}
79 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
83 1+(−0.939+1.62i)T+(−0.5−0.866i)T2 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2}
89 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
97 1+(0.326−0.118i)T+(0.766−0.642i)T2 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)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

−9.778811967377034943473399621945, −8.473349002577420088396669849758, −7.66869407674393974455403807767, −7.28968139882676662706049221602, −6.24065540826120027049745277935, −5.39521253728323000543857368702, −4.91672145825870559323984662226, −2.97927172552445241011113117541, −1.83582363627582254173468044454, −0.19785671053864246233162746748, 1.62327040332463009654822383384, 2.95612572687596962168003483353, 4.05474161767340251238105617188, 5.24428967078670367066662669490, 6.00531121013567481306194035779, 6.94656132317318540717616304132, 7.83051447252347100912182512786, 8.458515105810944955369685820120, 9.551844332534569212555842754037, 10.28121779564040163535543332555

Graph of the ZZ-function along the critical line