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Properties

Label 2-504-168.101-c1-0-1
Degree 22
Conductor 504504
Sign 0.266−0.963i0.266 - 0.963i
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  + (−0.169 − 1.40i)2-s + (−1.94 + 0.475i)4-s + (−1.53 + 0.887i)5-s + (0.843 − 2.50i)7-s + (0.995 + 2.64i)8-s + (1.50 + 2.00i)10-s + (−2.09 + 3.62i)11-s − 3.76·13-s + (−3.66 − 0.760i)14-s + (3.54 − 1.84i)16-s + (−2.32 + 4.02i)17-s + (0.0315 + 0.0545i)19-s + (2.56 − 2.45i)20-s + (5.44 + 2.32i)22-s + (−4.05 + 2.34i)23-s + ⋯
L(s)  = 1  + (−0.119 − 0.992i)2-s + (−0.971 + 0.237i)4-s + (−0.687 + 0.397i)5-s + (0.318 − 0.947i)7-s + (0.352 + 0.935i)8-s + (0.476 + 0.635i)10-s + (−0.631 + 1.09i)11-s − 1.04·13-s + (−0.979 − 0.203i)14-s + (0.887 − 0.461i)16-s + (−0.563 + 0.976i)17-s + (0.00723 + 0.0125i)19-s + (0.573 − 0.549i)20-s + (1.16 + 0.495i)22-s + (−0.845 + 0.488i)23-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((0.266−0.963i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s+1/2) L(s)=((0.266−0.963i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: 0.266−0.963i0.266 - 0.963i
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), 0.266−0.963i)(2,\ 504,\ (\ :1/2),\ 0.266 - 0.963i)

Particular Values

L(1)L(1) ≈\approx 0.329174+0.250489i0.329174 + 0.250489i
L(12)L(\frac12) ≈\approx 0.329174+0.250489i0.329174 + 0.250489i
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+(0.169+1.40i)T 1 + (0.169 + 1.40i)T
3 1 1
7 1+(−0.843+2.50i)T 1 + (-0.843 + 2.50i)T
good5 1+(1.53−0.887i)T+(2.5−4.33i)T2 1 + (1.53 - 0.887i)T + (2.5 - 4.33i)T^{2}
11 1+(2.09−3.62i)T+(−5.5−9.52i)T2 1 + (2.09 - 3.62i)T + (-5.5 - 9.52i)T^{2}
13 1+3.76T+13T2 1 + 3.76T + 13T^{2}
17 1+(2.32−4.02i)T+(−8.5−14.7i)T2 1 + (2.32 - 4.02i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.0315−0.0545i)T+(−9.5+16.4i)T2 1 + (-0.0315 - 0.0545i)T + (-9.5 + 16.4i)T^{2}
23 1+(4.05−2.34i)T+(11.5−19.9i)T2 1 + (4.05 - 2.34i)T + (11.5 - 19.9i)T^{2}
29 1−6.47T+29T2 1 - 6.47T + 29T^{2}
31 1+(−6.64−3.83i)T+(15.5+26.8i)T2 1 + (-6.64 - 3.83i)T + (15.5 + 26.8i)T^{2}
37 1+(2.91−1.68i)T+(18.5−32.0i)T2 1 + (2.91 - 1.68i)T + (18.5 - 32.0i)T^{2}
41 1+8.06T+41T2 1 + 8.06T + 41T^{2}
43 1−9.94iT−43T2 1 - 9.94iT - 43T^{2}
47 1+(0.338+0.586i)T+(−23.5+40.7i)T2 1 + (0.338 + 0.586i)T + (-23.5 + 40.7i)T^{2}
53 1+(−2.36+4.09i)T+(−26.5−45.8i)T2 1 + (-2.36 + 4.09i)T + (-26.5 - 45.8i)T^{2}
59 1+(6.53+3.77i)T+(29.5+51.0i)T2 1 + (6.53 + 3.77i)T + (29.5 + 51.0i)T^{2}
61 1+(7.23+12.5i)T+(−30.5+52.8i)T2 1 + (7.23 + 12.5i)T + (-30.5 + 52.8i)T^{2}
67 1+(−5.67−3.27i)T+(33.5+58.0i)T2 1 + (-5.67 - 3.27i)T + (33.5 + 58.0i)T^{2}
71 1−5.10iT−71T2 1 - 5.10iT - 71T^{2}
73 1+(−1.29−0.746i)T+(36.5+63.2i)T2 1 + (-1.29 - 0.746i)T + (36.5 + 63.2i)T^{2}
79 1+(5.99+10.3i)T+(−39.5+68.4i)T2 1 + (5.99 + 10.3i)T + (-39.5 + 68.4i)T^{2}
83 1−7.64iT−83T2 1 - 7.64iT - 83T^{2}
89 1+(8.97+15.5i)T+(−44.5+77.0i)T2 1 + (8.97 + 15.5i)T + (-44.5 + 77.0i)T^{2}
97 1−4.24iT−97T2 1 - 4.24iT - 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

−11.05121180600501332368240698740, −10.19647319228403920100238011376, −9.827680753849937619418434830346, −8.297757360659927250902837896977, −7.74237118901345772445338535671, −6.76100785189259957171902177948, −4.92691855535438117098481466155, −4.30696638220566709773347179962, −3.15181001479045793769174164274, −1.78598837912263658857963764715, 0.25318532133685993499727184827, 2.71306367703762849053460671869, 4.35683037582920791864205436528, 5.14224977503260599903476830056, 6.05714234249942402576504809967, 7.18166999691958839830108915654, 8.241025023240333738868021481219, 8.526435502779513169583516817548, 9.600785518460110222011874884178, 10.58731446930770975462594931468

Graph of the ZZ-function along the critical line