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Properties

Label 2-504-56.3-c1-0-5
Degree 22
Conductor 504504
Sign −0.0235−0.999i-0.0235 - 0.999i
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  + (−1.13 − 0.849i)2-s + (0.557 + 1.92i)4-s + (1.59 + 2.76i)5-s + (−0.694 + 2.55i)7-s + (1.00 − 2.64i)8-s + (0.542 − 4.48i)10-s + (−0.800 + 1.38i)11-s + 1.38·13-s + (2.95 − 2.29i)14-s + (−3.37 + 2.14i)16-s + (−3.48 − 2.01i)17-s + (−4.56 + 2.63i)19-s + (−4.42 + 4.61i)20-s + (2.08 − 0.888i)22-s + (−3.83 + 2.21i)23-s + ⋯
L(s)  = 1  + (−0.799 − 0.600i)2-s + (0.278 + 0.960i)4-s + (0.714 + 1.23i)5-s + (−0.262 + 0.964i)7-s + (0.353 − 0.935i)8-s + (0.171 − 1.41i)10-s + (−0.241 + 0.418i)11-s + 0.385·13-s + (0.789 − 0.614i)14-s + (−0.844 + 0.535i)16-s + (−0.845 − 0.488i)17-s + (−1.04 + 0.605i)19-s + (−0.988 + 1.03i)20-s + (0.444 − 0.189i)22-s + (−0.798 + 0.461i)23-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((−0.0235−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0235 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+1/2) L(s)=((−0.0235−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0235 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: −0.0235−0.999i-0.0235 - 0.999i
Analytic conductor: 4.024464.02446
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ504(451,⋅)\chi_{504} (451, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :1/2), −0.0235−0.999i)(2,\ 504,\ (\ :1/2),\ -0.0235 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.588231+0.602226i0.588231 + 0.602226i
L(12)L(\frac12) ≈\approx 0.588231+0.602226i0.588231 + 0.602226i
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+(1.13+0.849i)T 1 + (1.13 + 0.849i)T
3 1 1
7 1+(0.694−2.55i)T 1 + (0.694 - 2.55i)T
good5 1+(−1.59−2.76i)T+(−2.5+4.33i)T2 1 + (-1.59 - 2.76i)T + (-2.5 + 4.33i)T^{2}
11 1+(0.800−1.38i)T+(−5.5−9.52i)T2 1 + (0.800 - 1.38i)T + (-5.5 - 9.52i)T^{2}
13 1−1.38T+13T2 1 - 1.38T + 13T^{2}
17 1+(3.48+2.01i)T+(8.5+14.7i)T2 1 + (3.48 + 2.01i)T + (8.5 + 14.7i)T^{2}
19 1+(4.56−2.63i)T+(9.5−16.4i)T2 1 + (4.56 - 2.63i)T + (9.5 - 16.4i)T^{2}
23 1+(3.83−2.21i)T+(11.5−19.9i)T2 1 + (3.83 - 2.21i)T + (11.5 - 19.9i)T^{2}
29 1+5.10iT−29T2 1 + 5.10iT - 29T^{2}
31 1+(−0.0579+0.100i)T+(−15.5−26.8i)T2 1 + (-0.0579 + 0.100i)T + (-15.5 - 26.8i)T^{2}
37 1+(−4.63+2.67i)T+(18.5−32.0i)T2 1 + (-4.63 + 2.67i)T + (18.5 - 32.0i)T^{2}
41 1−4.21iT−41T2 1 - 4.21iT - 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+(−5.05−8.76i)T+(−23.5+40.7i)T2 1 + (-5.05 - 8.76i)T + (-23.5 + 40.7i)T^{2}
53 1+(−6.13−3.54i)T+(26.5+45.8i)T2 1 + (-6.13 - 3.54i)T + (26.5 + 45.8i)T^{2}
59 1+(−4.38−2.53i)T+(29.5+51.0i)T2 1 + (-4.38 - 2.53i)T + (29.5 + 51.0i)T^{2}
61 1+(−4.21−7.29i)T+(−30.5+52.8i)T2 1 + (-4.21 - 7.29i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.01+8.69i)T+(−33.5−58.0i)T2 1 + (-5.01 + 8.69i)T + (-33.5 - 58.0i)T^{2}
71 1−5.29iT−71T2 1 - 5.29iT - 71T^{2}
73 1+(−9.30−5.37i)T+(36.5+63.2i)T2 1 + (-9.30 - 5.37i)T + (36.5 + 63.2i)T^{2}
79 1+(10.3−5.96i)T+(39.5−68.4i)T2 1 + (10.3 - 5.96i)T + (39.5 - 68.4i)T^{2}
83 1+14.9iT−83T2 1 + 14.9iT - 83T^{2}
89 1+(1.5−0.866i)T+(44.5−77.0i)T2 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2}
97 1+2.87iT−97T2 1 + 2.87iT - 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

−11.02224663376967936118737337913, −10.17598647144315995550496416266, −9.579392015328024083227766509705, −8.665835491337125180621450540955, −7.64905744494781785715195881177, −6.58338029689523042415590875861, −5.91361952366068390920058635114, −4.09606281138815564564131487621, −2.70184217697430889809419679483, −2.12199384444095951122335904591, 0.62771475276367029318442108251, 2.01550256147075625645287827576, 4.17430848331995248055302044593, 5.20492832330551522113677853505, 6.21154729799317109659483924952, 6.99838291221786992074319592271, 8.353710331274755311052781841805, 8.699668173570238174941223466816, 9.699542012604970536442808308669, 10.48189005664696768832057620725

Graph of the ZZ-function along the critical line