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Properties

Label 2-12e2-144.11-c1-0-12
Degree 22
Conductor 144144
Sign 0.991−0.132i0.991 - 0.132i
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  + (−1.07 + 0.916i)2-s + (1.67 − 0.427i)3-s + (0.320 − 1.97i)4-s + (−0.170 − 0.0458i)5-s + (−1.41 + 1.99i)6-s + (1.17 − 2.03i)7-s + (1.46 + 2.42i)8-s + (2.63 − 1.43i)9-s + (0.226 − 0.107i)10-s + (−0.340 + 0.0913i)11-s + (−0.305 − 3.45i)12-s + (−1.49 − 0.399i)13-s + (0.598 + 3.26i)14-s + (−0.306 − 0.00382i)15-s + (−3.79 − 1.26i)16-s + 3.58i·17-s + ⋯
L(s)  = 1  + (−0.761 + 0.647i)2-s + (0.969 − 0.246i)3-s + (0.160 − 0.987i)4-s + (−0.0764 − 0.0204i)5-s + (−0.578 + 0.815i)6-s + (0.443 − 0.768i)7-s + (0.517 + 0.855i)8-s + (0.878 − 0.478i)9-s + (0.0715 − 0.0339i)10-s + (−0.102 + 0.0275i)11-s + (−0.0880 − 0.996i)12-s + (−0.413 − 0.110i)13-s + (0.160 + 0.873i)14-s + (−0.0791 − 0.000988i)15-s + (−0.948 − 0.316i)16-s + 0.868i·17-s + ⋯

Functional equation

Λ(s)=(144s/2 ΓC(s) L(s)=((0.991−0.132i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(144s/2 ΓC(s+1/2) L(s)=((0.991−0.132i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 144144    =    24⋅322^{4} \cdot 3^{2}
Sign: 0.991−0.132i0.991 - 0.132i
Analytic conductor: 1.149841.14984
Root analytic conductor: 1.072301.07230
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ144(11,⋅)\chi_{144} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 144, ( :1/2), 0.991−0.132i)(2,\ 144,\ (\ :1/2),\ 0.991 - 0.132i)

Particular Values

L(1)L(1) ≈\approx 1.05864+0.0703719i1.05864 + 0.0703719i
L(12)L(\frac12) ≈\approx 1.05864+0.0703719i1.05864 + 0.0703719i
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+(1.07−0.916i)T 1 + (1.07 - 0.916i)T
3 1+(−1.67+0.427i)T 1 + (-1.67 + 0.427i)T
good5 1+(0.170+0.0458i)T+(4.33+2.5i)T2 1 + (0.170 + 0.0458i)T + (4.33 + 2.5i)T^{2}
7 1+(−1.17+2.03i)T+(−3.5−6.06i)T2 1 + (-1.17 + 2.03i)T + (-3.5 - 6.06i)T^{2}
11 1+(0.340−0.0913i)T+(9.52−5.5i)T2 1 + (0.340 - 0.0913i)T + (9.52 - 5.5i)T^{2}
13 1+(1.49+0.399i)T+(11.2+6.5i)T2 1 + (1.49 + 0.399i)T + (11.2 + 6.5i)T^{2}
17 1−3.58iT−17T2 1 - 3.58iT - 17T^{2}
19 1+(−5.36−5.36i)T+19iT2 1 + (-5.36 - 5.36i)T + 19iT^{2}
23 1+(−0.165+0.0953i)T+(11.5−19.9i)T2 1 + (-0.165 + 0.0953i)T + (11.5 - 19.9i)T^{2}
29 1+(9.10−2.43i)T+(25.1−14.5i)T2 1 + (9.10 - 2.43i)T + (25.1 - 14.5i)T^{2}
31 1+(3.43−1.98i)T+(15.5−26.8i)T2 1 + (3.43 - 1.98i)T + (15.5 - 26.8i)T^{2}
37 1+(3.28+3.28i)T+37iT2 1 + (3.28 + 3.28i)T + 37iT^{2}
41 1+(4.25+7.37i)T+(−20.5+35.5i)T2 1 + (4.25 + 7.37i)T + (-20.5 + 35.5i)T^{2}
43 1+(−1.09−4.09i)T+(−37.2+21.5i)T2 1 + (-1.09 - 4.09i)T + (-37.2 + 21.5i)T^{2}
47 1+(−4.93+8.53i)T+(−23.5−40.7i)T2 1 + (-4.93 + 8.53i)T + (-23.5 - 40.7i)T^{2}
53 1+(4.83−4.83i)T−53iT2 1 + (4.83 - 4.83i)T - 53iT^{2}
59 1+(0.720−2.68i)T+(−51.0−29.5i)T2 1 + (0.720 - 2.68i)T + (-51.0 - 29.5i)T^{2}
61 1+(−2.13−7.97i)T+(−52.8+30.5i)T2 1 + (-2.13 - 7.97i)T + (-52.8 + 30.5i)T^{2}
67 1+(3.06−11.4i)T+(−58.0−33.5i)T2 1 + (3.06 - 11.4i)T + (-58.0 - 33.5i)T^{2}
71 1−1.13iT−71T2 1 - 1.13iT - 71T^{2}
73 1+5.67iT−73T2 1 + 5.67iT - 73T^{2}
79 1+(−12.8−7.42i)T+(39.5+68.4i)T2 1 + (-12.8 - 7.42i)T + (39.5 + 68.4i)T^{2}
83 1+(3.31+12.3i)T+(−71.8+41.5i)T2 1 + (3.31 + 12.3i)T + (-71.8 + 41.5i)T^{2}
89 1−3.05T+89T2 1 - 3.05T + 89T^{2}
97 1+(0.996−1.72i)T+(−48.5−84.0i)T2 1 + (0.996 - 1.72i)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

−13.50287055351023921067931358826, −12.15631270372261387263129552336, −10.67742863554048508224456721445, −9.868997182241823847297747865005, −8.797861062741152124770885230439, −7.73044010056297975789014413082, −7.24889233642238225958545710451, −5.62350172824571111955628803289, −3.88255319808369016891371714270, −1.70112376159042157314701772551, 2.12382356680289453530598186440, 3.35550892050438307896237433282, 4.97746858154872983542807247827, 7.23364026890578226585867385603, 8.003498210119450136105467790288, 9.277740402699432728755930817001, 9.550159359149155262550260104818, 11.08318753361510630391886489279, 11.83294173994983227124013242282, 13.05850556389694420787965853921

Graph of the ZZ-function along the critical line