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Properties

Label 2-12e2-144.13-c1-0-4
Degree 22
Conductor 144144
Sign −0.379−0.925i-0.379 - 0.925i
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  + (1.04 + 0.948i)2-s + (−1.13 + 1.31i)3-s + (0.199 + 1.98i)4-s + (0.00302 + 0.0112i)5-s + (−2.43 + 0.300i)6-s + (−1.05 + 0.610i)7-s + (−1.67 + 2.27i)8-s + (−0.435 − 2.96i)9-s + (−0.00753 + 0.0147i)10-s + (1.83 + 0.490i)11-s + (−2.83 − 1.99i)12-s + (5.06 − 1.35i)13-s + (−1.68 − 0.362i)14-s + (−0.0182 − 0.00881i)15-s + (−3.92 + 0.795i)16-s − 1.54·17-s + ⋯
L(s)  = 1  + (0.741 + 0.670i)2-s + (−0.653 + 0.756i)3-s + (0.0999 + 0.994i)4-s + (0.00135 + 0.00504i)5-s + (−0.992 + 0.122i)6-s + (−0.399 + 0.230i)7-s + (−0.593 + 0.804i)8-s + (−0.145 − 0.989i)9-s + (−0.00238 + 0.00465i)10-s + (0.551 + 0.147i)11-s + (−0.818 − 0.574i)12-s + (1.40 − 0.376i)13-s + (−0.451 − 0.0970i)14-s + (−0.00470 − 0.00227i)15-s + (−0.980 + 0.198i)16-s − 0.374·17-s + ⋯

Functional equation

Λ(s)=(144s/2 ΓC(s) L(s)=((−0.379−0.925i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(144s/2 ΓC(s+1/2) L(s)=((−0.379−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 144144    =    24⋅322^{4} \cdot 3^{2}
Sign: −0.379−0.925i-0.379 - 0.925i
Analytic conductor: 1.149841.14984
Root analytic conductor: 1.072301.07230
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ144(13,⋅)\chi_{144} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 144, ( :1/2), −0.379−0.925i)(2,\ 144,\ (\ :1/2),\ -0.379 - 0.925i)

Particular Values

L(1)L(1) ≈\approx 0.721276+1.07591i0.721276 + 1.07591i
L(12)L(\frac12) ≈\approx 0.721276+1.07591i0.721276 + 1.07591i
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+(−1.04−0.948i)T 1 + (-1.04 - 0.948i)T
3 1+(1.13−1.31i)T 1 + (1.13 - 1.31i)T
good5 1+(−0.00302−0.0112i)T+(−4.33+2.5i)T2 1 + (-0.00302 - 0.0112i)T + (-4.33 + 2.5i)T^{2}
7 1+(1.05−0.610i)T+(3.5−6.06i)T2 1 + (1.05 - 0.610i)T + (3.5 - 6.06i)T^{2}
11 1+(−1.83−0.490i)T+(9.52+5.5i)T2 1 + (-1.83 - 0.490i)T + (9.52 + 5.5i)T^{2}
13 1+(−5.06+1.35i)T+(11.2−6.5i)T2 1 + (-5.06 + 1.35i)T + (11.2 - 6.5i)T^{2}
17 1+1.54T+17T2 1 + 1.54T + 17T^{2}
19 1+(−4.06−4.06i)T+19iT2 1 + (-4.06 - 4.06i)T + 19iT^{2}
23 1+(5.20+3.00i)T+(11.5+19.9i)T2 1 + (5.20 + 3.00i)T + (11.5 + 19.9i)T^{2}
29 1+(−0.798+2.98i)T+(−25.1−14.5i)T2 1 + (-0.798 + 2.98i)T + (-25.1 - 14.5i)T^{2}
31 1+(−2.92+5.07i)T+(−15.5−26.8i)T2 1 + (-2.92 + 5.07i)T + (-15.5 - 26.8i)T^{2}
37 1+(−0.923+0.923i)T−37iT2 1 + (-0.923 + 0.923i)T - 37iT^{2}
41 1+(3.20+1.85i)T+(20.5+35.5i)T2 1 + (3.20 + 1.85i)T + (20.5 + 35.5i)T^{2}
43 1+(4.84+1.29i)T+(37.2+21.5i)T2 1 + (4.84 + 1.29i)T + (37.2 + 21.5i)T^{2}
47 1+(1.31+2.27i)T+(−23.5+40.7i)T2 1 + (1.31 + 2.27i)T + (-23.5 + 40.7i)T^{2}
53 1+(8.88−8.88i)T−53iT2 1 + (8.88 - 8.88i)T - 53iT^{2}
59 1+(−2.35−8.78i)T+(−51.0+29.5i)T2 1 + (-2.35 - 8.78i)T + (-51.0 + 29.5i)T^{2}
61 1+(−3.24+12.1i)T+(−52.8−30.5i)T2 1 + (-3.24 + 12.1i)T + (-52.8 - 30.5i)T^{2}
67 1+(11.8−3.18i)T+(58.0−33.5i)T2 1 + (11.8 - 3.18i)T + (58.0 - 33.5i)T^{2}
71 1+14.2iT−71T2 1 + 14.2iT - 71T^{2}
73 1−4.32iT−73T2 1 - 4.32iT - 73T^{2}
79 1+(0.261+0.453i)T+(−39.5+68.4i)T2 1 + (0.261 + 0.453i)T + (-39.5 + 68.4i)T^{2}
83 1+(−2.91+10.8i)T+(−71.8−41.5i)T2 1 + (-2.91 + 10.8i)T + (-71.8 - 41.5i)T^{2}
89 1−10.7iT−89T2 1 - 10.7iT - 89T^{2}
97 1+(−8.78−15.2i)T+(−48.5+84.0i)T2 1 + (-8.78 - 15.2i)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

−13.52145879843990760199229253939, −12.34690209515854284634536661300, −11.65401458380039595573840310815, −10.48197708543895387968390540602, −9.227494574013062126411398740612, −8.108053594063796241800326016155, −6.43602887372083957141686932922, −5.89025309558422299803301251706, −4.46335790020508748019976614957, −3.40021891608268219640152799252, 1.37937119597792766440501865919, 3.36057641070113924390476422480, 4.92729158963637660332567442475, 6.20333377200019114911762274731, 6.93582331065155197178678150889, 8.719408905625572990428553925704, 10.05648929861357108131815245791, 11.24529584389976008216714045658, 11.65806428269807100365027874102, 12.88061824634386302788117821528

Graph of the ZZ-function along the critical line