# Properties

 Label 2-126-63.58-c1-0-3 Degree $2$ Conductor $126$ Sign $0.381 + 0.924i$ Analytic cond. $1.00611$ Root an. cond. $1.00305$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + (−1.64 + 0.545i)3-s + 4-s + (−0.794 − 1.37i)5-s + (1.64 − 0.545i)6-s + (1.23 − 2.33i)7-s − 8-s + (2.40 − 1.79i)9-s + (0.794 + 1.37i)10-s + (0.794 − 1.37i)11-s + (−1.64 + 0.545i)12-s + (2.40 − 4.16i)13-s + (−1.23 + 2.33i)14-s + (2.05 + 1.82i)15-s + 16-s + (−2.69 − 4.67i)17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (−0.949 + 0.314i)3-s + 0.5·4-s + (−0.355 − 0.615i)5-s + (0.671 − 0.222i)6-s + (0.468 − 0.883i)7-s − 0.353·8-s + (0.801 − 0.597i)9-s + (0.251 + 0.434i)10-s + (0.239 − 0.414i)11-s + (−0.474 + 0.157i)12-s + (0.667 − 1.15i)13-s + (−0.331 + 0.624i)14-s + (0.530 + 0.472i)15-s + 0.250·16-s + (−0.654 − 1.13i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$126$$    =    $$2 \cdot 3^{2} \cdot 7$$ Sign: $0.381 + 0.924i$ Analytic conductor: $$1.00611$$ Root analytic conductor: $$1.00305$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{126} (121, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 126,\ (\ :1/2),\ 0.381 + 0.924i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.459259 - 0.307363i$$ $$L(\frac12)$$ $$\approx$$ $$0.459259 - 0.307363i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + (1.64 - 0.545i)T$$
7 $$1 + (-1.23 + 2.33i)T$$
good5 $$1 + (0.794 + 1.37i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-0.794 + 1.37i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 + (2.69 + 4.67i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (3.54 - 6.14i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (0.150 + 0.260i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-4.13 - 7.16i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + 2.71T + 31T^{2}$$
37 $$1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 - 2.66T + 47T^{2}$$
53 $$1 + (-2.44 - 4.23i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 - 6.47T + 59T^{2}$$
61 $$1 + 4.47T + 61T^{2}$$
67 $$1 + 10.0T + 67T^{2}$$
71 $$1 - 12.7T + 71T^{2}$$
73 $$1 + (-8.02 - 13.9i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 - 8.38T + 79T^{2}$$
83 $$1 + (-1.18 - 2.04i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-0.712 - 1.23i)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

−12.87484094954525194593663051091, −11.97082810644464250127551134373, −10.84732959791900234598302120257, −10.39670005005889916484766415281, −8.938586019222316464060645460913, −7.894191538808791936014529306411, −6.64343629647277429914394841098, −5.29011304663618827340426332134, −3.92107866396784158616243738204, −0.873986433272503384728038232416, 2.03750631602677229208003221670, 4.46608122365669305639871803248, 6.18356623052483934234293234208, 6.88601815630164515905870459924, 8.244284997232272599270867137285, 9.343063590666177901406344973970, 10.83955254478111696525142261941, 11.29599807704208171125261324161, 12.18388711779839018787106036372, 13.36279010228431881942096846163