Properties

Label 2-1920-1920.1589-c0-0-0
Degree 22
Conductor 19201920
Sign 0.9410.336i0.941 - 0.336i
Analytic cond. 0.9582040.958204
Root an. cond. 0.9788790.978879
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  + (−0.290 + 0.956i)2-s + (−0.0980 − 0.995i)3-s + (−0.831 − 0.555i)4-s + (0.881 + 0.471i)5-s + (0.980 + 0.195i)6-s + (0.773 − 0.634i)8-s + (−0.980 + 0.195i)9-s + (−0.707 + 0.707i)10-s + (−0.471 + 0.881i)12-s + (0.382 − 0.923i)15-s + (0.382 + 0.923i)16-s + (0.360 + 0.871i)17-s + (0.0980 − 0.995i)18-s + (0.448 − 1.47i)19-s + (−0.471 − 0.881i)20-s + ⋯
L(s)  = 1  + (−0.290 + 0.956i)2-s + (−0.0980 − 0.995i)3-s + (−0.831 − 0.555i)4-s + (0.881 + 0.471i)5-s + (0.980 + 0.195i)6-s + (0.773 − 0.634i)8-s + (−0.980 + 0.195i)9-s + (−0.707 + 0.707i)10-s + (−0.471 + 0.881i)12-s + (0.382 − 0.923i)15-s + (0.382 + 0.923i)16-s + (0.360 + 0.871i)17-s + (0.0980 − 0.995i)18-s + (0.448 − 1.47i)19-s + (−0.471 − 0.881i)20-s + ⋯

Functional equation

Λ(s)=(1920s/2ΓC(s)L(s)=((0.9410.336i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1920s/2ΓC(s)L(s)=((0.9410.336i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27352^{7} \cdot 3 \cdot 5
Sign: 0.9410.336i0.941 - 0.336i
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1920(1589,)\chi_{1920} (1589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1920, ( :0), 0.9410.336i)(2,\ 1920,\ (\ :0),\ 0.941 - 0.336i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.0427525451.042752545
L(12)L(\frac12) \approx 1.0427525451.042752545
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.2900.956i)T 1 + (0.290 - 0.956i)T
3 1+(0.0980+0.995i)T 1 + (0.0980 + 0.995i)T
5 1+(0.8810.471i)T 1 + (-0.881 - 0.471i)T
good7 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
11 1+(0.1950.980i)T2 1 + (0.195 - 0.980i)T^{2}
13 1+(0.5550.831i)T2 1 + (0.555 - 0.831i)T^{2}
17 1+(0.3600.871i)T+(0.707+0.707i)T2 1 + (-0.360 - 0.871i)T + (-0.707 + 0.707i)T^{2}
19 1+(0.448+1.47i)T+(0.8310.555i)T2 1 + (-0.448 + 1.47i)T + (-0.831 - 0.555i)T^{2}
23 1+(1.591.06i)T+(0.382+0.923i)T2 1 + (-1.59 - 1.06i)T + (0.382 + 0.923i)T^{2}
29 1+(0.1950.980i)T2 1 + (-0.195 - 0.980i)T^{2}
31 1+(1.17+1.17i)T+iT2 1 + (1.17 + 1.17i)T + iT^{2}
37 1+(0.831+0.555i)T2 1 + (-0.831 + 0.555i)T^{2}
41 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
43 1+(0.980+0.195i)T2 1 + (0.980 + 0.195i)T^{2}
47 1+(1.83+0.761i)T+(0.7070.707i)T2 1 + (-1.83 + 0.761i)T + (0.707 - 0.707i)T^{2}
53 1+(0.301+0.247i)T+(0.1950.980i)T2 1 + (-0.301 + 0.247i)T + (0.195 - 0.980i)T^{2}
59 1+(0.555+0.831i)T2 1 + (0.555 + 0.831i)T^{2}
61 1+(1.26+0.124i)T+(0.9800.195i)T2 1 + (-1.26 + 0.124i)T + (0.980 - 0.195i)T^{2}
67 1+(0.980+0.195i)T2 1 + (-0.980 + 0.195i)T^{2}
71 1+(0.9230.382i)T2 1 + (-0.923 - 0.382i)T^{2}
73 1+(0.9230.382i)T2 1 + (0.923 - 0.382i)T^{2}
79 1+(0.7070.292i)T+(0.707+0.707i)T2 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2}
83 1+(1.87+0.569i)T+(0.831+0.555i)T2 1 + (1.87 + 0.569i)T + (0.831 + 0.555i)T^{2}
89 1+(0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
97 1iT2 1 - iT^{2}
show more
show less
   L(s)=p j=12(1αj,pps)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

−9.163933745373036149093871193706, −8.677061245747683996535880742050, −7.50846344472881775201866301683, −7.16203717701029736901789038156, −6.40565218644316870527369658182, −5.60524600912997447131616002552, −5.11722866394464969010825842335, −3.54826147450151661954165612300, −2.30776179858977917781816227474, −1.13497901164359435096039850187, 1.18460624136507505350009319289, 2.55225176479048293361818252346, 3.35868147002420190777648456915, 4.39673322907077775785415046406, 5.16859284503079845222837618126, 5.73620210111394689392220766630, 7.08244962936253591615489374387, 8.240194235697336982575661041314, 8.943035410316241211890130633184, 9.425711979736080864640419452248

Graph of the ZZ-function along the critical line