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Properties

Label 2-12e2-144.11-c1-0-16
Degree $2$
Conductor $144$
Sign $0.768 + 0.639i$
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  + (0.688 − 1.23i)2-s + (1.10 + 1.33i)3-s + (−1.05 − 1.70i)4-s + (0.664 + 0.178i)5-s + (2.40 − 0.446i)6-s + (0.645 − 1.11i)7-s + (−2.82 + 0.129i)8-s + (−0.559 + 2.94i)9-s + (0.677 − 0.698i)10-s + (3.21 − 0.860i)11-s + (1.10 − 3.28i)12-s + (−4.74 − 1.27i)13-s + (−0.937 − 1.56i)14-s + (0.496 + 1.08i)15-s + (−1.78 + 3.57i)16-s + 5.58i·17-s + ⋯
L(s)  = 1  + (0.486 − 0.873i)2-s + (0.637 + 0.770i)3-s + (−0.526 − 0.850i)4-s + (0.297 + 0.0796i)5-s + (0.983 − 0.182i)6-s + (0.244 − 0.422i)7-s + (−0.998 + 0.0456i)8-s + (−0.186 + 0.982i)9-s + (0.214 − 0.220i)10-s + (0.968 − 0.259i)11-s + (0.319 − 0.947i)12-s + (−1.31 − 0.352i)13-s + (−0.250 − 0.418i)14-s + (0.128 + 0.279i)15-s + (−0.446 + 0.894i)16-s + 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.768 + 0.639i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.768 + 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49754 - 0.541483i\)
\(L(\frac12)\) \(\approx\) \(1.49754 - 0.541483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.688 + 1.23i)T \)
3 \( 1 + (-1.10 - 1.33i)T \)
good5 \( 1 + (-0.664 - 0.178i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.645 + 1.11i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.21 + 0.860i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.74 + 1.27i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 5.58iT - 17T^{2} \)
19 \( 1 + (2.49 + 2.49i)T + 19iT^{2} \)
23 \( 1 + (2.36 - 1.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.95 - 0.792i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-5.28 + 3.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.507 - 0.507i)T + 37iT^{2} \)
41 \( 1 + (-4.89 - 8.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.254 - 0.949i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-6.13 + 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.601 - 0.601i)T - 53iT^{2} \)
59 \( 1 + (-1.28 + 4.77i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.90 + 10.8i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.0295 - 0.110i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.0447iT - 71T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 + (-2.50 - 1.44i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.01 - 3.79i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (-4.41 + 7.63i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.09922210061007790673499406167, −11.92988062322418391378699187557, −10.83707118560808245556092244499, −10.02876946704864467798938227619, −9.239712029860852839445584968349, −7.996993520319502246536579363900, −6.13134889517005725772449292194, −4.68994559020438820420859011833, −3.74430714208859079142474023266, −2.20246958531668556763664102473, 2.45300818335388549102721879228, 4.21175919832984363082146953782, 5.70253716562732335634887128064, 6.88721959833471951344045816538, 7.67009910976264429980881239450, 8.915127787997026031688704716346, 9.595926219028649090517333322523, 11.93024414738925937920834358571, 12.22272049219969137843581302174, 13.48427112854635897038813299989

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