↑

Properties

Label 2-504-63.16-c1-0-9
Degree 22
Conductor 504504
Sign 0.643−0.765i0.643 - 0.765i
Analytic cond. 4.024464.02446
Root an. cond. 2.006102.00610
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  + (−1.40 + 1.01i)3-s + 3.84·5-s + (0.676 + 2.55i)7-s + (0.953 − 2.84i)9-s + 1.80·11-s + (−0.692 − 1.19i)13-s + (−5.40 + 3.88i)15-s + (−0.833 − 1.44i)17-s + (−0.0802 + 0.138i)19-s + (−3.53 − 2.91i)21-s + 3.20·23-s + 9.75·25-s + (1.53 + 4.96i)27-s + (−3.78 + 6.54i)29-s + (−1.61 + 2.78i)31-s + ⋯
L(s)  = 1  + (−0.811 + 0.584i)3-s + 1.71·5-s + (0.255 + 0.966i)7-s + (0.317 − 0.948i)9-s + 0.544·11-s + (−0.192 − 0.332i)13-s + (−1.39 + 1.00i)15-s + (−0.202 − 0.350i)17-s + (−0.0184 + 0.0318i)19-s + (−0.772 − 0.635i)21-s + 0.667·23-s + 1.95·25-s + (0.295 + 0.955i)27-s + (−0.701 + 1.21i)29-s + (−0.289 + 0.500i)31-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((0.643−0.765i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+1/2) L(s)=((0.643−0.765i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: 0.643−0.765i0.643 - 0.765i
Analytic conductor: 4.024464.02446
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ504(457,⋅)\chi_{504} (457, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :1/2), 0.643−0.765i)(2,\ 504,\ (\ :1/2),\ 0.643 - 0.765i)

Particular Values

L(1)L(1) ≈\approx 1.39037+0.647046i1.39037 + 0.647046i
L(12)L(\frac12) ≈\approx 1.39037+0.647046i1.39037 + 0.647046i
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+(1.40−1.01i)T 1 + (1.40 - 1.01i)T
7 1+(−0.676−2.55i)T 1 + (-0.676 - 2.55i)T
good5 1−3.84T+5T2 1 - 3.84T + 5T^{2}
11 1−1.80T+11T2 1 - 1.80T + 11T^{2}
13 1+(0.692+1.19i)T+(−6.5+11.2i)T2 1 + (0.692 + 1.19i)T + (-6.5 + 11.2i)T^{2}
17 1+(0.833+1.44i)T+(−8.5+14.7i)T2 1 + (0.833 + 1.44i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.0802−0.138i)T+(−9.5−16.4i)T2 1 + (0.0802 - 0.138i)T + (-9.5 - 16.4i)T^{2}
23 1−3.20T+23T2 1 - 3.20T + 23T^{2}
29 1+(3.78−6.54i)T+(−14.5−25.1i)T2 1 + (3.78 - 6.54i)T + (-14.5 - 25.1i)T^{2}
31 1+(1.61−2.78i)T+(−15.5−26.8i)T2 1 + (1.61 - 2.78i)T + (-15.5 - 26.8i)T^{2}
37 1+(−1.58+2.74i)T+(−18.5−32.0i)T2 1 + (-1.58 + 2.74i)T + (-18.5 - 32.0i)T^{2}
41 1+(−6.00−10.3i)T+(−20.5+35.5i)T2 1 + (-6.00 - 10.3i)T + (-20.5 + 35.5i)T^{2}
43 1+(−3.45+5.98i)T+(−21.5−37.2i)T2 1 + (-3.45 + 5.98i)T + (-21.5 - 37.2i)T^{2}
47 1+(5.71+9.90i)T+(−23.5+40.7i)T2 1 + (5.71 + 9.90i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.37−2.38i)T+(−26.5+45.8i)T2 1 + (-1.37 - 2.38i)T + (-26.5 + 45.8i)T^{2}
59 1+(7.53−13.0i)T+(−29.5−51.0i)T2 1 + (7.53 - 13.0i)T + (-29.5 - 51.0i)T^{2}
61 1+(−4.60−7.96i)T+(−30.5+52.8i)T2 1 + (-4.60 - 7.96i)T + (-30.5 + 52.8i)T^{2}
67 1+(−6.16+10.6i)T+(−33.5−58.0i)T2 1 + (-6.16 + 10.6i)T + (-33.5 - 58.0i)T^{2}
71 1+6.93T+71T2 1 + 6.93T + 71T^{2}
73 1+(6.22+10.7i)T+(−36.5+63.2i)T2 1 + (6.22 + 10.7i)T + (-36.5 + 63.2i)T^{2}
79 1+(8.03+13.9i)T+(−39.5+68.4i)T2 1 + (8.03 + 13.9i)T + (-39.5 + 68.4i)T^{2}
83 1+(1.45−2.51i)T+(−41.5−71.8i)T2 1 + (1.45 - 2.51i)T + (-41.5 - 71.8i)T^{2}
89 1+(−5.04+8.73i)T+(−44.5−77.0i)T2 1 + (-5.04 + 8.73i)T + (-44.5 - 77.0i)T^{2}
97 1+(−4.18+7.25i)T+(−48.5−84.0i)T2 1 + (-4.18 + 7.25i)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

−10.90135954527524634914616329232, −10.17546933122065857243629394986, −9.213314656216958133487151506135, −8.952066453788774060402109403781, −7.09221973557845614970021182640, −6.08837384213536255725632658061, −5.52936020995091292345988918560, −4.72793504412673048047904097407, −2.98659860882191519621250600529, −1.58962396303074821694752084904, 1.20717093618910807321623518352, 2.26476034277108504524132064076, 4.26345854055238837512642820556, 5.35462617561607626592055624742, 6.22663492472924622371332876568, 6.87050840427494049130761487756, 7.904087570184520968953435812321, 9.305356414489593547075518775236, 9.943685348718405237321151291999, 10.87275246214999305600427201191

Graph of the ZZ-function along the critical line