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Properties

Label 2-12e2-144.133-c1-0-18
Degree 22
Conductor 144144
Sign 0.461+0.887i0.461 + 0.887i
Analytic cond. 1.149841.14984
Root an. cond. 1.072301.07230
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  + (−0.366 + 1.36i)2-s − 1.73i·3-s + (−1.73 − i)4-s + (0.5 − 1.86i)5-s + (2.36 + 0.633i)6-s + (−3.86 − 2.23i)7-s + (2 − 1.99i)8-s − 2.99·9-s + (2.36 + 1.36i)10-s + (1.86 − 0.5i)11-s + (−1.73 + 2.99i)12-s + (2.23 + 0.598i)13-s + (4.46 − 4.46i)14-s + (−3.23 − 0.866i)15-s + (1.99 + 3.46i)16-s + 4·17-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s − 0.999i·3-s + (−0.866 − 0.5i)4-s + (0.223 − 0.834i)5-s + (0.965 + 0.258i)6-s + (−1.46 − 0.843i)7-s + (0.707 − 0.707i)8-s − 0.999·9-s + (0.748 + 0.431i)10-s + (0.562 − 0.150i)11-s + (−0.499 + 0.866i)12-s + (0.619 + 0.165i)13-s + (1.19 − 1.19i)14-s + (−0.834 − 0.223i)15-s + (0.499 + 0.866i)16-s + 0.970·17-s + ⋯

Functional equation

Λ(s)=(144s/2 ΓC(s) L(s)=((0.461+0.887i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(144s/2 ΓC(s+1/2) L(s)=((0.461+0.887i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 144144    =    24⋅322^{4} \cdot 3^{2}
Sign: 0.461+0.887i0.461 + 0.887i
Analytic conductor: 1.149841.14984
Root analytic conductor: 1.072301.07230
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ144(133,⋅)\chi_{144} (133, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 144, ( :1/2), 0.461+0.887i)(2,\ 144,\ (\ :1/2),\ 0.461 + 0.887i)

Particular Values

L(1)L(1) ≈\approx 0.663983−0.402914i0.663983 - 0.402914i
L(12)L(\frac12) ≈\approx 0.663983−0.402914i0.663983 - 0.402914i
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+(0.366−1.36i)T 1 + (0.366 - 1.36i)T
3 1+1.73iT 1 + 1.73iT
good5 1+(−0.5+1.86i)T+(−4.33−2.5i)T2 1 + (-0.5 + 1.86i)T + (-4.33 - 2.5i)T^{2}
7 1+(3.86+2.23i)T+(3.5+6.06i)T2 1 + (3.86 + 2.23i)T + (3.5 + 6.06i)T^{2}
11 1+(−1.86+0.5i)T+(9.52−5.5i)T2 1 + (-1.86 + 0.5i)T + (9.52 - 5.5i)T^{2}
13 1+(−2.23−0.598i)T+(11.2+6.5i)T2 1 + (-2.23 - 0.598i)T + (11.2 + 6.5i)T^{2}
17 1−4T+17T2 1 - 4T + 17T^{2}
19 1+(3−3i)T−19iT2 1 + (3 - 3i)T - 19iT^{2}
23 1+(−5.59+3.23i)T+(11.5−19.9i)T2 1 + (-5.59 + 3.23i)T + (11.5 - 19.9i)T^{2}
29 1+(−0.232−0.866i)T+(−25.1+14.5i)T2 1 + (-0.232 - 0.866i)T + (-25.1 + 14.5i)T^{2}
31 1+(4.59+7.96i)T+(−15.5+26.8i)T2 1 + (4.59 + 7.96i)T + (-15.5 + 26.8i)T^{2}
37 1+(4.26+4.26i)T+37iT2 1 + (4.26 + 4.26i)T + 37iT^{2}
41 1+(0.696−0.401i)T+(20.5−35.5i)T2 1 + (0.696 - 0.401i)T + (20.5 - 35.5i)T^{2}
43 1+(−6.33+1.69i)T+(37.2−21.5i)T2 1 + (-6.33 + 1.69i)T + (37.2 - 21.5i)T^{2}
47 1+(0.598−1.03i)T+(−23.5−40.7i)T2 1 + (0.598 - 1.03i)T + (-23.5 - 40.7i)T^{2}
53 1+(−5.73−5.73i)T+53iT2 1 + (-5.73 - 5.73i)T + 53iT^{2}
59 1+(−0.401+1.5i)T+(−51.0−29.5i)T2 1 + (-0.401 + 1.5i)T + (-51.0 - 29.5i)T^{2}
61 1+(−0.571−2.13i)T+(−52.8+30.5i)T2 1 + (-0.571 - 2.13i)T + (-52.8 + 30.5i)T^{2}
67 1+(−8.33−2.23i)T+(58.0+33.5i)T2 1 + (-8.33 - 2.23i)T + (58.0 + 33.5i)T^{2}
71 1−2.92iT−71T2 1 - 2.92iT - 71T^{2}
73 1+7.46iT−73T2 1 + 7.46iT - 73T^{2}
79 1+(0.866−1.5i)T+(−39.5−68.4i)T2 1 + (0.866 - 1.5i)T + (-39.5 - 68.4i)T^{2}
83 1+(−3.79−14.1i)T+(−71.8+41.5i)T2 1 + (-3.79 - 14.1i)T + (-71.8 + 41.5i)T^{2}
89 1+15.8iT−89T2 1 + 15.8iT - 89T^{2}
97 1+(0.5−0.866i)T+(−48.5−84.0i)T2 1 + (0.5 - 0.866i)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

−12.99867465693642473685032678544, −12.51674886113895300814304979273, −10.70885153941083645267889106240, −9.431577443043981254048006040540, −8.674970061269965289316890143279, −7.42532181144775920698853247253, −6.52813017654069927913816182951, −5.65611816523331335781761941739, −3.80806570891253224534312301153, −0.913121273494331531416946209800, 2.86516634859788755610611831208, 3.58274782711945581778145147432, 5.36116064237471455353744662941, 6.70775926826863527110384410604, 8.699291742906490160131586632283, 9.377429014192716184212136941679, 10.24018892755385668166855631815, 11.01899735195603900907673627857, 12.10597555208484007716368641119, 13.07732750753171736091682875281

Graph of the ZZ-function along the critical line