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Properties

Label 2-12e2-144.13-c1-0-13
Degree 22
Conductor 144144
Sign −0.999+0.0436i-0.999 + 0.0436i
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−1.5 + 0.866i)3-s + (1.73 + i)4-s + (−0.267 − i)5-s + (2.36 − 0.633i)6-s + (−2.36 + 1.36i)7-s + (−1.99 − 2i)8-s + (1.5 − 2.59i)9-s + 1.46i·10-s + (−4.23 − 1.13i)11-s − 3.46·12-s + (−3.36 + 0.901i)13-s + (3.73 − 0.999i)14-s + (1.26 + 1.26i)15-s + (1.99 + 3.46i)16-s − 5.73·17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.866 + 0.499i)3-s + (0.866 + 0.5i)4-s + (−0.119 − 0.447i)5-s + (0.965 − 0.258i)6-s + (−0.894 + 0.516i)7-s + (−0.707 − 0.707i)8-s + (0.5 − 0.866i)9-s + 0.462i·10-s + (−1.27 − 0.341i)11-s − 0.999·12-s + (−0.933 + 0.250i)13-s + (0.997 − 0.267i)14-s + (0.327 + 0.327i)15-s + (0.499 + 0.866i)16-s − 1.39·17-s + ⋯

Functional equation

Λ(s)=(144s/2 ΓC(s) L(s)=((−0.999+0.0436i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0436i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(144s/2 ΓC(s+1/2) L(s)=((−0.999+0.0436i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0436i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 144144    =    24⋅322^{4} \cdot 3^{2}
Sign: −0.999+0.0436i-0.999 + 0.0436i
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: 11
Selberg data: (2, 144, ( :1/2), −0.999+0.0436i)(2,\ 144,\ (\ :1/2),\ -0.999 + 0.0436i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+(1.36+0.366i)T 1 + (1.36 + 0.366i)T
3 1+(1.5−0.866i)T 1 + (1.5 - 0.866i)T
good5 1+(0.267+i)T+(−4.33+2.5i)T2 1 + (0.267 + i)T + (-4.33 + 2.5i)T^{2}
7 1+(2.36−1.36i)T+(3.5−6.06i)T2 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2}
11 1+(4.23+1.13i)T+(9.52+5.5i)T2 1 + (4.23 + 1.13i)T + (9.52 + 5.5i)T^{2}
13 1+(3.36−0.901i)T+(11.2−6.5i)T2 1 + (3.36 - 0.901i)T + (11.2 - 6.5i)T^{2}
17 1+5.73T+17T2 1 + 5.73T + 17T^{2}
19 1+(2.36+2.36i)T+19iT2 1 + (2.36 + 2.36i)T + 19iT^{2}
23 1+(−4.09−2.36i)T+(11.5+19.9i)T2 1 + (-4.09 - 2.36i)T + (11.5 + 19.9i)T^{2}
29 1+(0.633−2.36i)T+(−25.1−14.5i)T2 1 + (0.633 - 2.36i)T + (-25.1 - 14.5i)T^{2}
31 1+(0.267−0.464i)T+(−15.5−26.8i)T2 1 + (0.267 - 0.464i)T + (-15.5 - 26.8i)T^{2}
37 1+(−4.73+4.73i)T−37iT2 1 + (-4.73 + 4.73i)T - 37iT^{2}
41 1+(−2.59−1.5i)T+(20.5+35.5i)T2 1 + (-2.59 - 1.5i)T + (20.5 + 35.5i)T^{2}
43 1+(8.33+2.23i)T+(37.2+21.5i)T2 1 + (8.33 + 2.23i)T + (37.2 + 21.5i)T^{2}
47 1+(−3.83−6.63i)T+(−23.5+40.7i)T2 1 + (-3.83 - 6.63i)T + (-23.5 + 40.7i)T^{2}
53 1+(7.46−7.46i)T−53iT2 1 + (7.46 - 7.46i)T - 53iT^{2}
59 1+(1.96+7.33i)T+(−51.0+29.5i)T2 1 + (1.96 + 7.33i)T + (-51.0 + 29.5i)T^{2}
61 1+(3−11.1i)T+(−52.8−30.5i)T2 1 + (3 - 11.1i)T + (-52.8 - 30.5i)T^{2}
67 1+(−6.59+1.76i)T+(58.0−33.5i)T2 1 + (-6.59 + 1.76i)T + (58.0 - 33.5i)T^{2}
71 1−2.92iT−71T2 1 - 2.92iT - 71T^{2}
73 1+6.26iT−73T2 1 + 6.26iT - 73T^{2}
79 1+(6+10.3i)T+(−39.5+68.4i)T2 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2}
83 1+(−0.366+1.36i)T+(−71.8−41.5i)T2 1 + (-0.366 + 1.36i)T + (-71.8 - 41.5i)T^{2}
89 1+2iT−89T2 1 + 2iT - 89T^{2}
97 1+(5.86+10.1i)T+(−48.5+84.0i)T2 1 + (5.86 + 10.1i)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

−12.54018850008471894709126549214, −11.29987979499578249116801848688, −10.59849603715670775130457849865, −9.493053514051915962514721438516, −8.803182800278576681178790141765, −7.24309662101516330627775025940, −6.15713347037993703936757882100, −4.74503044809173584197382618871, −2.79700404624231203986260100087, 0, 2.50903389517737029009421037721, 5.03899878026064847581236989605, 6.50269972909689479506041912865, 7.08865245648435220077871456057, 8.139057893690085841848224931074, 9.766870970907271383751969584107, 10.51747393909866444324423786089, 11.23282193544176049839975743328, 12.57458115351867995140311781522

Graph of the ZZ-function along the critical line