↑

Properties

Label 2-504-168.101-c1-0-18
Degree 22
Conductor 504504
Sign 0.881+0.472i0.881 + 0.472i
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 − 0.127i)2-s + (1.96 + 0.359i)4-s + (1.87 − 1.08i)5-s + (2.55 + 0.669i)7-s + (−2.72 − 0.758i)8-s + (−2.77 + 1.28i)10-s + (1.67 − 2.89i)11-s + 1.74·13-s + (−3.51 − 1.27i)14-s + (3.74 + 1.41i)16-s + (−0.283 + 0.491i)17-s + (−0.270 − 0.467i)19-s + (4.07 − 1.45i)20-s + (−2.72 + 3.86i)22-s + (−5.21 + 3.01i)23-s + ⋯
L(s)  = 1  + (−0.995 − 0.0903i)2-s + (0.983 + 0.179i)4-s + (0.837 − 0.483i)5-s + (0.967 + 0.253i)7-s + (−0.963 − 0.268i)8-s + (−0.877 + 0.405i)10-s + (0.503 − 0.872i)11-s + 0.483·13-s + (−0.940 − 0.339i)14-s + (0.935 + 0.354i)16-s + (−0.0688 + 0.119i)17-s + (−0.0619 − 0.107i)19-s + (0.910 − 0.324i)20-s + (−0.580 + 0.823i)22-s + (−1.08 + 0.627i)23-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((0.881+0.472i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+1/2) L(s)=((0.881+0.472i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: 0.881+0.472i0.881 + 0.472i
Analytic conductor: 4.024464.02446
Root analytic conductor: 2.006102.00610
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ504(269,⋅)\chi_{504} (269, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :1/2), 0.881+0.472i)(2,\ 504,\ (\ :1/2),\ 0.881 + 0.472i)

Particular Values

L(1)L(1) ≈\approx 1.18217−0.297126i1.18217 - 0.297126i
L(12)L(\frac12) ≈\approx 1.18217−0.297126i1.18217 - 0.297126i
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+(1.40+0.127i)T 1 + (1.40 + 0.127i)T
3 1 1
7 1+(−2.55−0.669i)T 1 + (-2.55 - 0.669i)T
good5 1+(−1.87+1.08i)T+(2.5−4.33i)T2 1 + (-1.87 + 1.08i)T + (2.5 - 4.33i)T^{2}
11 1+(−1.67+2.89i)T+(−5.5−9.52i)T2 1 + (-1.67 + 2.89i)T + (-5.5 - 9.52i)T^{2}
13 1−1.74T+13T2 1 - 1.74T + 13T^{2}
17 1+(0.283−0.491i)T+(−8.5−14.7i)T2 1 + (0.283 - 0.491i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.270+0.467i)T+(−9.5+16.4i)T2 1 + (0.270 + 0.467i)T + (-9.5 + 16.4i)T^{2}
23 1+(5.21−3.01i)T+(11.5−19.9i)T2 1 + (5.21 - 3.01i)T + (11.5 - 19.9i)T^{2}
29 1−1.77T+29T2 1 - 1.77T + 29T^{2}
31 1+(6.56+3.79i)T+(15.5+26.8i)T2 1 + (6.56 + 3.79i)T + (15.5 + 26.8i)T^{2}
37 1+(−9.60+5.54i)T+(18.5−32.0i)T2 1 + (-9.60 + 5.54i)T + (18.5 - 32.0i)T^{2}
41 1−7.77T+41T2 1 - 7.77T + 41T^{2}
43 1−1.80iT−43T2 1 - 1.80iT - 43T^{2}
47 1+(−0.679−1.17i)T+(−23.5+40.7i)T2 1 + (-0.679 - 1.17i)T + (-23.5 + 40.7i)T^{2}
53 1+(−1.46+2.54i)T+(−26.5−45.8i)T2 1 + (-1.46 + 2.54i)T + (-26.5 - 45.8i)T^{2}
59 1+(−9.84−5.68i)T+(29.5+51.0i)T2 1 + (-9.84 - 5.68i)T + (29.5 + 51.0i)T^{2}
61 1+(5.60+9.71i)T+(−30.5+52.8i)T2 1 + (5.60 + 9.71i)T + (-30.5 + 52.8i)T^{2}
67 1+(10.7+6.21i)T+(33.5+58.0i)T2 1 + (10.7 + 6.21i)T + (33.5 + 58.0i)T^{2}
71 1−7.79iT−71T2 1 - 7.79iT - 71T^{2}
73 1+(−7.56−4.36i)T+(36.5+63.2i)T2 1 + (-7.56 - 4.36i)T + (36.5 + 63.2i)T^{2}
79 1+(4.77+8.27i)T+(−39.5+68.4i)T2 1 + (4.77 + 8.27i)T + (-39.5 + 68.4i)T^{2}
83 1−15.9iT−83T2 1 - 15.9iT - 83T^{2}
89 1+(6.30+10.9i)T+(−44.5+77.0i)T2 1 + (6.30 + 10.9i)T + (-44.5 + 77.0i)T^{2}
97 1−16.4iT−97T2 1 - 16.4iT - 97T^{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

−10.97749760017023383327709798976, −9.713404718850945562050821655329, −9.120996071343508100867969064502, −8.322050786808555257658407551004, −7.53237643263802216040970657624, −6.10600761486083659750138581644, −5.60958808487587870396104085719, −3.94056628900772601888058826694, −2.29237409246094589423192807395, −1.21477577274253163169828381047, 1.48743169964490963109449858606, 2.48294234592600286457171470297, 4.25437052265798851608917101221, 5.69771840658342174857394443914, 6.55109668541690386311842245128, 7.45622736118693583522110370446, 8.337254880331970242355134294595, 9.285757820769499726954715179971, 10.11018214959266827209341129152, 10.72864386639742602945674830570

Graph of the ZZ-function along the critical line