↑

Properties

Label 2-3240-360.149-c0-0-11
Degree 22
Conductor 32403240
Sign −0.984+0.173i-0.984 + 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

Related objects

Downloads

Learn more

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+0.173i-0.984 + 0.173i
Analytic conductor: 1.616971.61697
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3240(1349,⋅)\chi_{3240} (1349, \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 0.65550246940.6555024694
L(12)L(\frac12) ≈\approx 0.65550246940.6555024694
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}
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

−8.536995632400850000115093830754, −8.186921106169253930510932042461, −7.09740091834331479366786297305, −6.25646073826014993535611414877, −5.16384300675554929154509548313, −4.56780495873831782646466257111, −3.69247992007079660168033112125, −2.50195402673133539025340630028, −1.85718105529215349721674279729, −0.44841016151069281578945038242, 1.62701644104950975271333890849, 2.61451903850541710074882236799, 3.74451570577425587629406938305, 5.10179098569669817114853987564, 5.29391189981654398968661497272, 6.42039318673403441028219751063, 7.11382867356695623465213506854, 7.38973233283220143214869712989, 8.453653159677570187174207579985, 9.067042929703856636288672322902

Graph of the ZZ-function along the critical line