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Properties

Label 2-3024-252.103-c1-0-8
Degree 22
Conductor 30243024
Sign 0.932−0.361i0.932 - 0.361i
Analytic cond. 24.146724.1467
Root an. cond. 4.913934.91393
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  + (−1.53 + 0.886i)5-s + (−2.64 − 0.0987i)7-s + (−5.33 − 3.08i)11-s + (−4.60 − 2.65i)13-s + (−4.69 + 2.71i)17-s + (0.935 − 1.62i)19-s + (0.562 − 0.324i)23-s + (−0.928 + 1.60i)25-s + (1.14 + 1.98i)29-s + 10.2·31-s + (4.14 − 2.19i)35-s + (1.09 − 1.89i)37-s + (6.64 + 3.83i)41-s + (−1.07 + 0.620i)43-s + 0.468·47-s + ⋯
L(s)  = 1  + (−0.686 + 0.396i)5-s + (−0.999 − 0.0373i)7-s + (−1.60 − 0.929i)11-s + (−1.27 − 0.737i)13-s + (−1.13 + 0.657i)17-s + (0.214 − 0.371i)19-s + (0.117 − 0.0677i)23-s + (−0.185 + 0.321i)25-s + (0.213 + 0.369i)29-s + 1.83·31-s + (0.700 − 0.370i)35-s + (0.180 − 0.311i)37-s + (1.03 + 0.599i)41-s + (−0.163 + 0.0946i)43-s + 0.0683·47-s + ⋯

Functional equation

Λ(s)=(3024s/2 ΓC(s) L(s)=((0.932−0.361i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3024s/2 ΓC(s+1/2) L(s)=((0.932−0.361i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30243024    =    24⋅33⋅72^{4} \cdot 3^{3} \cdot 7
Sign: 0.932−0.361i0.932 - 0.361i
Analytic conductor: 24.146724.1467
Root analytic conductor: 4.913934.91393
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3024(1279,⋅)\chi_{3024} (1279, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3024, ( :1/2), 0.932−0.361i)(2,\ 3024,\ (\ :1/2),\ 0.932 - 0.361i)

Particular Values

L(1)L(1) ≈\approx 0.60864290890.6086429089
L(12)L(\frac12) ≈\approx 0.60864290890.6086429089
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
7 1+(2.64+0.0987i)T 1 + (2.64 + 0.0987i)T
good5 1+(1.53−0.886i)T+(2.5−4.33i)T2 1 + (1.53 - 0.886i)T + (2.5 - 4.33i)T^{2}
11 1+(5.33+3.08i)T+(5.5+9.52i)T2 1 + (5.33 + 3.08i)T + (5.5 + 9.52i)T^{2}
13 1+(4.60+2.65i)T+(6.5+11.2i)T2 1 + (4.60 + 2.65i)T + (6.5 + 11.2i)T^{2}
17 1+(4.69−2.71i)T+(8.5−14.7i)T2 1 + (4.69 - 2.71i)T + (8.5 - 14.7i)T^{2}
19 1+(−0.935+1.62i)T+(−9.5−16.4i)T2 1 + (-0.935 + 1.62i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.562+0.324i)T+(11.5−19.9i)T2 1 + (-0.562 + 0.324i)T + (11.5 - 19.9i)T^{2}
29 1+(−1.14−1.98i)T+(−14.5+25.1i)T2 1 + (-1.14 - 1.98i)T + (-14.5 + 25.1i)T^{2}
31 1−10.2T+31T2 1 - 10.2T + 31T^{2}
37 1+(−1.09+1.89i)T+(−18.5−32.0i)T2 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2}
41 1+(−6.64−3.83i)T+(20.5+35.5i)T2 1 + (-6.64 - 3.83i)T + (20.5 + 35.5i)T^{2}
43 1+(1.07−0.620i)T+(21.5−37.2i)T2 1 + (1.07 - 0.620i)T + (21.5 - 37.2i)T^{2}
47 1−0.468T+47T2 1 - 0.468T + 47T^{2}
53 1+(−0.941−1.63i)T+(−26.5+45.8i)T2 1 + (-0.941 - 1.63i)T + (-26.5 + 45.8i)T^{2}
59 1−9.24T+59T2 1 - 9.24T + 59T^{2}
61 1+6.33iT−61T2 1 + 6.33iT - 61T^{2}
67 1−9.93iT−67T2 1 - 9.93iT - 67T^{2}
71 1+11.1iT−71T2 1 + 11.1iT - 71T^{2}
73 1+(4.77−2.75i)T+(36.5−63.2i)T2 1 + (4.77 - 2.75i)T + (36.5 - 63.2i)T^{2}
79 1−10.5iT−79T2 1 - 10.5iT - 79T^{2}
83 1+(4.06+7.03i)T+(−41.5+71.8i)T2 1 + (4.06 + 7.03i)T + (-41.5 + 71.8i)T^{2}
89 1+(−0.228−0.132i)T+(44.5+77.0i)T2 1 + (-0.228 - 0.132i)T + (44.5 + 77.0i)T^{2}
97 1+(12.5−7.26i)T+(48.5−84.0i)T2 1 + (12.5 - 7.26i)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

−8.603935125061451664772461310294, −7.979509790710904397026015094783, −7.33015232723815699259598071330, −6.57058739769086968950006434149, −5.73175547159679148079240684625, −4.93383699140421930196658559902, −3.97290186568778175550096027899, −2.82808114084507696544226741579, −2.69086286936565058744525277408, −0.50534400096554261240341129702, 0.38351496065811961921914856634, 2.33266912278815929527665996158, 2.77501669060320854075488916213, 4.21508139244432880124233376040, 4.63723014828331485090267658784, 5.50032212716550735640005567744, 6.61597598664476062314711177297, 7.20374557188469089764026763755, 7.85667673976992426227105558123, 8.626623684462106473954743054361

Graph of the ZZ-function along the critical line