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Properties

Label 2-504-63.4-c1-0-5
Degree 22
Conductor 504504
Sign 0.593−0.805i0.593 - 0.805i
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  + (−1.21 + 1.22i)3-s − 0.481·5-s + (2.53 − 0.763i)7-s + (−0.0248 − 2.99i)9-s + 3.38·11-s + (−2.86 + 4.95i)13-s + (0.587 − 0.592i)15-s + (2.75 − 4.77i)17-s + (2.18 + 3.77i)19-s + (−2.15 + 4.04i)21-s + 3.62·23-s − 4.76·25-s + (3.71 + 3.62i)27-s + (1.53 + 2.65i)29-s + (4.67 + 8.09i)31-s + ⋯
L(s)  = 1  + (−0.704 + 0.710i)3-s − 0.215·5-s + (0.957 − 0.288i)7-s + (−0.00827 − 0.999i)9-s + 1.01·11-s + (−0.793 + 1.37i)13-s + (0.151 − 0.152i)15-s + (0.668 − 1.15i)17-s + (0.500 + 0.866i)19-s + (−0.469 + 0.882i)21-s + 0.756·23-s − 0.953·25-s + (0.715 + 0.698i)27-s + (0.284 + 0.492i)29-s + (0.839 + 1.45i)31-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((0.593−0.805i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+1/2) L(s)=((0.593−0.805i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: 0.593−0.805i0.593 - 0.805i
Analytic conductor: 4.024464.02446
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ504(193,⋅)\chi_{504} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :1/2), 0.593−0.805i)(2,\ 504,\ (\ :1/2),\ 0.593 - 0.805i)

Particular Values

L(1)L(1) ≈\approx 1.10037+0.555970i1.10037 + 0.555970i
L(12)L(\frac12) ≈\approx 1.10037+0.555970i1.10037 + 0.555970i
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+(1.21−1.22i)T 1 + (1.21 - 1.22i)T
7 1+(−2.53+0.763i)T 1 + (-2.53 + 0.763i)T
good5 1+0.481T+5T2 1 + 0.481T + 5T^{2}
11 1−3.38T+11T2 1 - 3.38T + 11T^{2}
13 1+(2.86−4.95i)T+(−6.5−11.2i)T2 1 + (2.86 - 4.95i)T + (-6.5 - 11.2i)T^{2}
17 1+(−2.75+4.77i)T+(−8.5−14.7i)T2 1 + (-2.75 + 4.77i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.18−3.77i)T+(−9.5+16.4i)T2 1 + (-2.18 - 3.77i)T + (-9.5 + 16.4i)T^{2}
23 1−3.62T+23T2 1 - 3.62T + 23T^{2}
29 1+(−1.53−2.65i)T+(−14.5+25.1i)T2 1 + (-1.53 - 2.65i)T + (-14.5 + 25.1i)T^{2}
31 1+(−4.67−8.09i)T+(−15.5+26.8i)T2 1 + (-4.67 - 8.09i)T + (-15.5 + 26.8i)T^{2}
37 1+(−1.48−2.57i)T+(−18.5+32.0i)T2 1 + (-1.48 - 2.57i)T + (-18.5 + 32.0i)T^{2}
41 1+(6.29−10.9i)T+(−20.5−35.5i)T2 1 + (6.29 - 10.9i)T + (-20.5 - 35.5i)T^{2}
43 1+(−1.90−3.30i)T+(−21.5+37.2i)T2 1 + (-1.90 - 3.30i)T + (-21.5 + 37.2i)T^{2}
47 1+(−1.88+3.26i)T+(−23.5−40.7i)T2 1 + (-1.88 + 3.26i)T + (-23.5 - 40.7i)T^{2}
53 1+(−5.57+9.66i)T+(−26.5−45.8i)T2 1 + (-5.57 + 9.66i)T + (-26.5 - 45.8i)T^{2}
59 1+(4.21+7.29i)T+(−29.5+51.0i)T2 1 + (4.21 + 7.29i)T + (-29.5 + 51.0i)T^{2}
61 1+(−3.64+6.31i)T+(−30.5−52.8i)T2 1 + (-3.64 + 6.31i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.28+2.22i)T+(−33.5+58.0i)T2 1 + (1.28 + 2.22i)T + (-33.5 + 58.0i)T^{2}
71 1+3.94T+71T2 1 + 3.94T + 71T^{2}
73 1+(0.862−1.49i)T+(−36.5−63.2i)T2 1 + (0.862 - 1.49i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.79+4.84i)T+(−39.5−68.4i)T2 1 + (-2.79 + 4.84i)T + (-39.5 - 68.4i)T^{2}
83 1+(0.119+0.206i)T+(−41.5+71.8i)T2 1 + (0.119 + 0.206i)T + (-41.5 + 71.8i)T^{2}
89 1+(−0.648−1.12i)T+(−44.5+77.0i)T2 1 + (-0.648 - 1.12i)T + (-44.5 + 77.0i)T^{2}
97 1+(7.02+12.1i)T+(−48.5+84.0i)T2 1 + (7.02 + 12.1i)T + (-48.5 + 84.0i)T^{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

−11.33147546251289551765032504121, −10.05211937282289036015427278174, −9.523412637174555157148309431953, −8.483630480535081273419686827173, −7.26179041016808669705392711878, −6.49571263249785033473144436843, −5.10405428660720499682558260837, −4.56497476214757147299284486265, −3.41677887481579616131045325736, −1.36816456744947388170719951542, 0.985915758282882827765597647637, 2.44862837209468926098984589226, 4.16756464279804003666447553276, 5.34221109628679268278903637713, 6.00098803802663380157183189475, 7.32424336412280455515859666420, 7.84549833377342961875956217047, 8.838810934930701107393336690848, 10.13162765219220704724359639114, 10.92099254243463127234786649848

Graph of the ZZ-function along the critical line