↑

Properties

Label 2-2160-15.14-c0-0-4
Degree 22
Conductor 21602160
Sign 0.707+0.707i0.707 + 0.707i
Analytic cond. 1.077981.07798
Root an. cond. 1.038251.03825
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  + (0.707 − 0.707i)5-s + i·7-s − 1.41i·11-s − i·13-s − 19-s + 1.41·23-s − 1.00i·25-s − 1.41i·29-s + (0.707 + 0.707i)35-s + i·37-s + 1.41i·41-s + 1.41·53-s + (−1.00 − 1.00i)55-s − 61-s + (−0.707 − 0.707i)65-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + i·7-s − 1.41i·11-s − i·13-s − 19-s + 1.41·23-s − 1.00i·25-s − 1.41i·29-s + (0.707 + 0.707i)35-s + i·37-s + 1.41i·41-s + 1.41·53-s + (−1.00 − 1.00i)55-s − 61-s + (−0.707 − 0.707i)65-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s) L(s)=((0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 0.707+0.707i0.707 + 0.707i
Analytic conductor: 1.077981.07798
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2160(1889,⋅)\chi_{2160} (1889, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2160, ( :0), 0.707+0.707i)(2,\ 2160,\ (\ :0),\ 0.707 + 0.707i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3077667341.307766734
L(12)L(\frac12) ≈\approx 1.3077667341.307766734
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 1
5 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good7 1−iT−T2 1 - iT - T^{2}
11 1+1.41iT−T2 1 + 1.41iT - T^{2}
13 1+iT−T2 1 + iT - T^{2}
17 1+T2 1 + T^{2}
19 1+T+T2 1 + T + T^{2}
23 1−1.41T+T2 1 - 1.41T + T^{2}
29 1+1.41iT−T2 1 + 1.41iT - T^{2}
31 1+T2 1 + T^{2}
37 1−iT−T2 1 - iT - T^{2}
41 1−1.41iT−T2 1 - 1.41iT - T^{2}
43 1−T2 1 - T^{2}
47 1+T2 1 + T^{2}
53 1−1.41T+T2 1 - 1.41T + T^{2}
59 1−T2 1 - T^{2}
61 1+T+T2 1 + T + T^{2}
67 1−iT−T2 1 - iT - T^{2}
71 1+1.41iT−T2 1 + 1.41iT - T^{2}
73 1−iT−T2 1 - iT - T^{2}
79 1−T+T2 1 - T + T^{2}
83 1+1.41T+T2 1 + 1.41T + T^{2}
89 1−T2 1 - T^{2}
97 1−iT−T2 1 - iT - 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

−8.981005944189793950412343155257, −8.527560263549555993759251617078, −7.937389993510061633114299284302, −6.54434918050619773763203943153, −5.89068762861157760291461271730, −5.38412833681878083567992811931, −4.46596650223500158213644188886, −3.13525931027800871266299781335, −2.40825720303266386130029002400, −0.994737123430730438835313106031, 1.58062512804029687945387432507, 2.43612157846312220841865952203, 3.68493575279538663600998276755, 4.47748150253517346951882482002, 5.33888141925683638682176174584, 6.50347751916732354917210509194, 7.09867980042917947804189051546, 7.38168266003385136569102654178, 8.859292724376237293166252724572, 9.308065633744256526859794573429

Graph of the ZZ-function along the critical line