↑

Properties

Label 2-3267-27.11-c0-0-0
Degree 22
Conductor 32673267
Sign −0.396+0.918i-0.396 + 0.918i
Analytic cond. 1.630441.63044
Root an. cond. 1.276881.27688
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.673 + 1.85i)5-s + (0.173 − 0.984i)9-s + (−0.766 − 0.642i)12-s + (−0.673 − 1.85i)15-s + (−0.939 + 0.342i)16-s + (−1.93 − 0.342i)20-s + (−1.70 + 0.300i)23-s + (−2.20 − 1.85i)25-s + (0.500 + 0.866i)27-s + (−0.266 − 1.50i)31-s + 36-s + (0.173 + 0.300i)37-s + (1.70 + 0.984i)45-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.673 + 1.85i)5-s + (0.173 − 0.984i)9-s + (−0.766 − 0.642i)12-s + (−0.673 − 1.85i)15-s + (−0.939 + 0.342i)16-s + (−1.93 − 0.342i)20-s + (−1.70 + 0.300i)23-s + (−2.20 − 1.85i)25-s + (0.500 + 0.866i)27-s + (−0.266 − 1.50i)31-s + 36-s + (0.173 + 0.300i)37-s + (1.70 + 0.984i)45-s + ⋯

Functional equation

Λ(s)=(3267s/2 ΓC(s) L(s)=((−0.396+0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((−0.396+0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: −0.396+0.918i-0.396 + 0.918i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(848,⋅)\chi_{3267} (848, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), −0.396+0.918i)(2,\ 3267,\ (\ :0),\ -0.396 + 0.918i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.41417411190.4141741119
L(12)L(\frac12) ≈\approx 0.41417411190.4141741119
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)
bad3 1+(0.766−0.642i)T 1 + (0.766 - 0.642i)T
11 1 1
good2 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
5 1+(0.673−1.85i)T+(−0.766−0.642i)T2 1 + (0.673 - 1.85i)T + (-0.766 - 0.642i)T^{2}
7 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
13 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
23 1+(1.70−0.300i)T+(0.939−0.342i)T2 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2}
29 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
31 1+(0.266+1.50i)T+(−0.939+0.342i)T2 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2}
37 1+(−0.173−0.300i)T+(−0.5+0.866i)T2 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2}
41 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
43 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
47 1+(−0.673−0.118i)T+(0.939+0.342i)T2 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2}
53 1−1.28iT−T2 1 - 1.28iT - T^{2}
59 1+(0.439−1.20i)T+(−0.766−0.642i)T2 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2}
61 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
67 1+(1.17−0.984i)T+(0.173−0.984i)T2 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2}
71 1+(−0.592+0.342i)T+(0.5−0.866i)T2 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2}
73 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
79 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
83 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
89 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
97 1+(−1.76+0.642i)T+(0.766−0.642i)T2 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2}
show more
show less
   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

−9.495648896886032071416134327814, −8.463513630631696319323001484642, −7.43730051076813037369158896490, −7.37735689023851735278019200913, −6.22354924734362051816494541778, −5.92802653320564818552787770780, −4.27736987807645307424078805091, −3.98175038357704093492069605889, −3.12769073541937506830247325078, −2.34089851591561749872558152667, 0.28202531224940203087612249285, 1.31079925555514298866661564220, 2.06246410010373805515962438385, 3.90952031118968737506283365658, 4.73545157711655159260924214821, 5.27905137393718108484681144092, 5.88223317118208659174833851725, 6.72222024629403536965585664842, 7.62715773457759367506961474012, 8.285232498524436230619196719861

Graph of the ZZ-function along the critical line