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Properties

Label 2-12e2-144.13-c1-0-15
Degree $2$
Conductor $144$
Sign $0.764 + 0.644i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.174 + 1.40i)2-s + (0.215 − 1.71i)3-s + (−1.93 − 0.489i)4-s + (−0.733 − 2.73i)5-s + (2.37 + 0.602i)6-s + (1.14 − 0.660i)7-s + (1.02 − 2.63i)8-s + (−2.90 − 0.740i)9-s + (3.96 − 0.551i)10-s + (1.28 + 0.343i)11-s + (−1.25 + 3.22i)12-s + (3.36 − 0.902i)13-s + (0.727 + 1.72i)14-s + (−4.86 + 0.670i)15-s + (3.51 + 1.90i)16-s − 7.60·17-s + ⋯
L(s)  = 1  + (−0.123 + 0.992i)2-s + (0.124 − 0.992i)3-s + (−0.969 − 0.244i)4-s + (−0.328 − 1.22i)5-s + (0.969 + 0.245i)6-s + (0.432 − 0.249i)7-s + (0.362 − 0.931i)8-s + (−0.969 − 0.246i)9-s + (1.25 − 0.174i)10-s + (0.387 + 0.103i)11-s + (−0.363 + 0.931i)12-s + (0.933 − 0.250i)13-s + (0.194 + 0.460i)14-s + (−1.25 + 0.173i)15-s + (0.879 + 0.475i)16-s − 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.764 + 0.644i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889200 - 0.325031i\)
\(L(\frac12)\) \(\approx\) \(0.889200 - 0.325031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.174 - 1.40i)T \)
3 \( 1 + (-0.215 + 1.71i)T \)
good5 \( 1 + (0.733 + 2.73i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.14 + 0.660i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.28 - 0.343i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.36 + 0.902i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 7.60T + 17T^{2} \)
19 \( 1 + (-4.32 - 4.32i)T + 19iT^{2} \)
23 \( 1 + (-3.46 - 1.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.950 + 3.54i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.569 + 0.985i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.26 + 2.26i)T - 37iT^{2} \)
41 \( 1 + (-1.42 - 0.821i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.17 + 1.65i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.58 - 7.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.72 + 7.72i)T - 53iT^{2} \)
59 \( 1 + (-1.28 - 4.80i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.66 - 9.92i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-13.9 + 3.73i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.87iT - 71T^{2} \)
73 \( 1 + 0.577iT - 73T^{2} \)
79 \( 1 + (-0.716 - 1.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.885 - 3.30i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 16.2iT - 89T^{2} \)
97 \( 1 + (0.648 + 1.12i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.27289829893982995997060777250, −12.29657714988539057695023987059, −11.15444553065583622888341465842, −9.311463909822391851265525810590, −8.521090127778163990451415360006, −7.79501147614648334858954224654, −6.61503518039986111715206072007, −5.44094227576619380441327136713, −4.12939385121960908393064631823, −1.13754062336198186371887200753, 2.65220811412533319378174478660, 3.77270666657025582533389161220, 4.98201147353581936154108863751, 6.77442271636791724979300122914, 8.494946520015291400322793665802, 9.187889715265563676305659719458, 10.46090043090364820896358330605, 11.22867636625127798277973680212, 11.52803012199233354941601907202, 13.31810204146894942382897396284

Graph of the $Z$-function along the critical line