↑

Properties

Label 2-1620-9.4-c1-0-9
Degree 22
Conductor 16201620
Sign 0.642+0.766i0.642 + 0.766i
Analytic cond. 12.935712.9357
Root an. cond. 3.596633.59663
Motivic weight 11
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)5-s + (−0.366 − 0.633i)7-s + (0.866 + 1.5i)11-s + (0.732 − 1.26i)13-s − 1.26·17-s + 2.46·19-s + (1.73 − 3i)23-s + (−0.499 − 0.866i)25-s + (2.13 + 3.69i)29-s + (3.96 − 6.86i)31-s − 0.732·35-s + 4.19·37-s + (0.401 − 0.696i)41-s + (−3.36 − 5.83i)43-s + (2.36 + 4.09i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.138 − 0.239i)7-s + (0.261 + 0.452i)11-s + (0.203 − 0.351i)13-s − 0.307·17-s + 0.565·19-s + (0.361 − 0.625i)23-s + (−0.0999 − 0.173i)25-s + (0.396 + 0.686i)29-s + (0.711 − 1.23i)31-s − 0.123·35-s + 0.689·37-s + (0.0627 − 0.108i)41-s + (−0.513 − 0.889i)43-s + (0.345 + 0.597i)47-s + ⋯

Functional equation

Λ(s)=(1620s/2 ΓC(s) L(s)=((0.642+0.766i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1620s/2 ΓC(s+1/2) L(s)=((0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16201620    =    22⋅34⋅52^{2} \cdot 3^{4} \cdot 5
Sign: 0.642+0.766i0.642 + 0.766i
Analytic conductor: 12.935712.9357
Root analytic conductor: 3.596633.59663
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1620(1081,⋅)\chi_{1620} (1081, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1620, ( :1/2), 0.642+0.766i)(2,\ 1620,\ (\ :1/2),\ 0.642 + 0.766i)

Particular Values

L(1)L(1) ≈\approx 1.7372246231.737224623
L(12)L(\frac12) ≈\approx 1.7372246231.737224623
L(32)L(\frac{3}{2}) 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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good7 1+(0.366+0.633i)T+(−3.5+6.06i)T2 1 + (0.366 + 0.633i)T + (-3.5 + 6.06i)T^{2}
11 1+(−0.866−1.5i)T+(−5.5+9.52i)T2 1 + (-0.866 - 1.5i)T + (-5.5 + 9.52i)T^{2}
13 1+(−0.732+1.26i)T+(−6.5−11.2i)T2 1 + (-0.732 + 1.26i)T + (-6.5 - 11.2i)T^{2}
17 1+1.26T+17T2 1 + 1.26T + 17T^{2}
19 1−2.46T+19T2 1 - 2.46T + 19T^{2}
23 1+(−1.73+3i)T+(−11.5−19.9i)T2 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.13−3.69i)T+(−14.5+25.1i)T2 1 + (-2.13 - 3.69i)T + (-14.5 + 25.1i)T^{2}
31 1+(−3.96+6.86i)T+(−15.5−26.8i)T2 1 + (-3.96 + 6.86i)T + (-15.5 - 26.8i)T^{2}
37 1−4.19T+37T2 1 - 4.19T + 37T^{2}
41 1+(−0.401+0.696i)T+(−20.5−35.5i)T2 1 + (-0.401 + 0.696i)T + (-20.5 - 35.5i)T^{2}
43 1+(3.36+5.83i)T+(−21.5+37.2i)T2 1 + (3.36 + 5.83i)T + (-21.5 + 37.2i)T^{2}
47 1+(−2.36−4.09i)T+(−23.5+40.7i)T2 1 + (-2.36 - 4.09i)T + (-23.5 + 40.7i)T^{2}
53 1+10.7T+53T2 1 + 10.7T + 53T^{2}
59 1+(−2.13+3.69i)T+(−29.5−51.0i)T2 1 + (-2.13 + 3.69i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2−3.46i)T+(−30.5+52.8i)T2 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2}
67 1+(−7.19+12.4i)T+(−33.5−58.0i)T2 1 + (-7.19 + 12.4i)T + (-33.5 - 58.0i)T^{2}
71 1+0.803T+71T2 1 + 0.803T + 71T^{2}
73 1−10.1T+73T2 1 - 10.1T + 73T^{2}
79 1+(3.19+5.53i)T+(−39.5+68.4i)T2 1 + (3.19 + 5.53i)T + (-39.5 + 68.4i)T^{2}
83 1+(4.56+7.90i)T+(−41.5+71.8i)T2 1 + (4.56 + 7.90i)T + (-41.5 + 71.8i)T^{2}
89 1+5.19T+89T2 1 + 5.19T + 89T^{2}
97 1+(−1.36−2.36i)T+(−48.5+84.0i)T2 1 + (-1.36 - 2.36i)T + (-48.5 + 84.0i)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

−9.341691004926548433649776213320, −8.515799349846804794505756762891, −7.73597881156011754866160024298, −6.83875699169215445204378109869, −6.08642069652963399029927411949, −5.09143726527632559171599928767, −4.33582107847498803953399312772, −3.27167056965582197922978959604, −2.10526354074546352033617889543, −0.77484711188084090977357244211, 1.21733416237829936893073573232, 2.57229822579732747643466573371, 3.42883372336940212003047841801, 4.50566069069556445041736967464, 5.50738025480898550302218151210, 6.32942161225038323282827338987, 6.97681224019572989391786094236, 7.970957091270820498142075215082, 8.747316199043512435041299928404, 9.537945349303752683151114247292

Graph of the ZZ-function along the critical line