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Properties

Label 2-504-168.5-c1-0-5
Degree 22
Conductor 504504
Sign 0.999+0.0285i0.999 + 0.0285i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (−2.87 − 1.65i)5-s + (2.5 + 0.866i)7-s − 2.82i·8-s + (2.34 + 4.06i)10-s + (2.87 + 4.97i)11-s − 4.69·13-s + (−2.44 − 2.82i)14-s + (−2.00 + 3.46i)16-s + (1.22 + 2.12i)17-s + (2.34 − 4.06i)19-s − 6.63i·20-s − 8.12i·22-s + (1.22 + 0.707i)23-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (0.499 + 0.866i)4-s + (−1.28 − 0.741i)5-s + (0.944 + 0.327i)7-s − 0.999i·8-s + (0.741 + 1.28i)10-s + (0.866 + 1.50i)11-s − 1.30·13-s + (−0.654 − 0.755i)14-s + (−0.500 + 0.866i)16-s + (0.297 + 0.514i)17-s + (0.538 − 0.931i)19-s − 1.48i·20-s − 1.73i·22-s + (0.255 + 0.147i)23-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((0.999+0.0285i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+1/2) L(s)=((0.999+0.0285i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0285i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: 0.999+0.0285i0.999 + 0.0285i
Analytic conductor: 4.024464.02446
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ504(341,⋅)\chi_{504} (341, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :1/2), 0.999+0.0285i)(2,\ 504,\ (\ :1/2),\ 0.999 + 0.0285i)

Particular Values

L(1)L(1) ≈\approx 0.822876−0.0117291i0.822876 - 0.0117291i
L(12)L(\frac12) ≈\approx 0.822876−0.0117291i0.822876 - 0.0117291i
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+(1.22+0.707i)T 1 + (1.22 + 0.707i)T
3 1 1
7 1+(−2.5−0.866i)T 1 + (-2.5 - 0.866i)T
good5 1+(2.87+1.65i)T+(2.5+4.33i)T2 1 + (2.87 + 1.65i)T + (2.5 + 4.33i)T^{2}
11 1+(−2.87−4.97i)T+(−5.5+9.52i)T2 1 + (-2.87 - 4.97i)T + (-5.5 + 9.52i)T^{2}
13 1+4.69T+13T2 1 + 4.69T + 13T^{2}
17 1+(−1.22−2.12i)T+(−8.5+14.7i)T2 1 + (-1.22 - 2.12i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.34+4.06i)T+(−9.5−16.4i)T2 1 + (-2.34 + 4.06i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.22−0.707i)T+(11.5+19.9i)T2 1 + (-1.22 - 0.707i)T + (11.5 + 19.9i)T^{2}
29 1−5.74T+29T2 1 - 5.74T + 29T^{2}
31 1+(−4.5+2.59i)T+(15.5−26.8i)T2 1 + (-4.5 + 2.59i)T + (15.5 - 26.8i)T^{2}
37 1+(−7.03−4.06i)T+(18.5+32.0i)T2 1 + (-7.03 - 4.06i)T + (18.5 + 32.0i)T^{2}
41 1−9.79T+41T2 1 - 9.79T + 41T^{2}
43 1−8.12iT−43T2 1 - 8.12iT - 43T^{2}
47 1+(3.67−6.36i)T+(−23.5−40.7i)T2 1 + (3.67 - 6.36i)T + (-23.5 - 40.7i)T^{2}
53 1+(2.87+4.97i)T+(−26.5+45.8i)T2 1 + (2.87 + 4.97i)T + (-26.5 + 45.8i)T^{2}
59 1+(−2.87+1.65i)T+(29.5−51.0i)T2 1 + (-2.87 + 1.65i)T + (29.5 - 51.0i)T^{2}
61 1+(−2.34+4.06i)T+(−30.5−52.8i)T2 1 + (-2.34 + 4.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(33.5−58.0i)T2 1 + (33.5 - 58.0i)T^{2}
71 1+1.41iT−71T2 1 + 1.41iT - 71T^{2}
73 1+(3−1.73i)T+(36.5−63.2i)T2 1 + (3 - 1.73i)T + (36.5 - 63.2i)T^{2}
79 1+(−3.5+6.06i)T+(−39.5−68.4i)T2 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2}
83 1−3.31iT−83T2 1 - 3.31iT - 83T^{2}
89 1+(4.89−8.48i)T+(−44.5−77.0i)T2 1 + (4.89 - 8.48i)T + (-44.5 - 77.0i)T^{2}
97 1−1.73iT−97T2 1 - 1.73iT - 97T^{2}
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   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

−11.12904325984422283425212641841, −9.782390367338062443730789481077, −9.264987279654103152555301292183, −8.081018069702570658364956901354, −7.73993356308687919649677580398, −6.75728123597318013600533391308, −4.73341736071739413864982616358, −4.31682890311997059742172817592, −2.62782326270394987387772824794, −1.15987594951749495044191543573, 0.836239326669811585426799495341, 2.83576822472231661314703626669, 4.18738986204643616292712871544, 5.45846220430211546809866322046, 6.67205706674191253499150518809, 7.52348295859090884703434695187, 8.037138237342509920427442830652, 8.922076360355870096559829613115, 10.10283060153389360272528899839, 10.90477669685708528667316231044

Graph of the ZZ-function along the critical line