# Properties

 Label 2-12e2-144.13-c1-0-11 Degree $2$ Conductor $144$ Sign $0.999 - 0.00740i$ Analytic cond. $1.14984$ Root an. cond. $1.07230$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (1.30 + 0.550i)2-s + (−1.21 − 1.22i)3-s + (1.39 + 1.43i)4-s + (−0.0468 − 0.174i)5-s + (−0.912 − 2.27i)6-s + (4.04 − 2.33i)7-s + (1.02 + 2.63i)8-s + (−0.0241 + 2.99i)9-s + (0.0351 − 0.253i)10-s + (−0.598 − 0.160i)11-s + (0.0627 − 3.46i)12-s + (−4.41 + 1.18i)13-s + (6.55 − 0.816i)14-s + (−0.157 + 0.270i)15-s + (−0.112 + 3.99i)16-s − 4.34·17-s + ⋯
 L(s)  = 1 + (0.921 + 0.389i)2-s + (−0.704 − 0.709i)3-s + (0.697 + 0.716i)4-s + (−0.0209 − 0.0781i)5-s + (−0.372 − 0.928i)6-s + (1.52 − 0.883i)7-s + (0.363 + 0.931i)8-s + (−0.00806 + 0.999i)9-s + (0.0111 − 0.0801i)10-s + (−0.180 − 0.0483i)11-s + (0.0181 − 0.999i)12-s + (−1.22 + 0.327i)13-s + (1.75 − 0.218i)14-s + (−0.0407 + 0.0698i)15-s + (−0.0281 + 0.999i)16-s − 1.05·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00740i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00740i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$144$$    =    $$2^{4} \cdot 3^{2}$$ Sign: $0.999 - 0.00740i$ 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: $$(2,\ 144,\ (\ :1/2),\ 0.999 - 0.00740i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.57736 + 0.00584189i$$ $$L(\frac12)$$ $$\approx$$ $$1.57736 + 0.00584189i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.30 - 0.550i)T$$
3 $$1 + (1.21 + 1.22i)T$$
good5 $$1 + (0.0468 + 0.174i)T + (-4.33 + 2.5i)T^{2}$$
7 $$1 + (-4.04 + 2.33i)T + (3.5 - 6.06i)T^{2}$$
11 $$1 + (0.598 + 0.160i)T + (9.52 + 5.5i)T^{2}$$
13 $$1 + (4.41 - 1.18i)T + (11.2 - 6.5i)T^{2}$$
17 $$1 + 4.34T + 17T^{2}$$
19 $$1 + (1.23 + 1.23i)T + 19iT^{2}$$
23 $$1 + (3.86 + 2.23i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (2.31 - 8.64i)T + (-25.1 - 14.5i)T^{2}$$
31 $$1 + (2.25 - 3.90i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-2.79 + 2.79i)T - 37iT^{2}$$
41 $$1 + (-3.67 - 2.12i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (0.0131 + 0.00351i)T + (37.2 + 21.5i)T^{2}$$
47 $$1 + (1.17 + 2.03i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (0.519 - 0.519i)T - 53iT^{2}$$
59 $$1 + (2.95 + 11.0i)T + (-51.0 + 29.5i)T^{2}$$
61 $$1 + (-0.588 + 2.19i)T + (-52.8 - 30.5i)T^{2}$$
67 $$1 + (-7.04 + 1.88i)T + (58.0 - 33.5i)T^{2}$$
71 $$1 - 7.55iT - 71T^{2}$$
73 $$1 - 2.92iT - 73T^{2}$$
79 $$1 + (-1.45 - 2.52i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (-1.99 + 7.44i)T + (-71.8 - 41.5i)T^{2}$$
89 $$1 + 3.18iT - 89T^{2}$$
97 $$1 + (-8.03 - 13.9i)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

−13.04642067087922961059518050148, −12.27129845839619483392344482032, −11.22386597053567400382754203918, −10.68971146899647538609201255257, −8.416210636972137644528021066069, −7.43618634346110570970907194521, −6.70533612440639539807757082415, −5.11906212096217091705261345568, −4.50128645339276521769818344427, −2.06713649045134293165188693029, 2.30662752398928025661270380539, 4.29541722014113673135693635292, 5.11484786636068878271642663689, 6.04309828302893764346248748506, 7.66010837273689396981938343316, 9.267162933548278284646144893396, 10.40890897228331980456503602961, 11.37663297117618852698862547668, 11.84366945716299976523407466131, 12.87954378993050905782257114099