# Properties

 Label 2-126-63.20-c1-0-2 Degree $2$ Conductor $126$ Sign $0.341 - 0.940i$ 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 + (0.866 + 0.5i)2-s + (0.0967 + 1.72i)3-s + (0.499 + 0.866i)4-s + (0.183 + 0.317i)5-s + (−0.780 + 1.54i)6-s + (−0.624 − 2.57i)7-s + 0.999i·8-s + (−2.98 + 0.334i)9-s + 0.366i·10-s + (0.579 + 0.334i)11-s + (−1.44 + 0.948i)12-s + (0.867 − 0.500i)13-s + (0.744 − 2.53i)14-s + (−0.531 + 0.347i)15-s + (−0.5 + 0.866i)16-s + 4.98·17-s + ⋯
 L(s)  = 1 + (0.612 + 0.353i)2-s + (0.0558 + 0.998i)3-s + (0.249 + 0.433i)4-s + (0.0819 + 0.141i)5-s + (−0.318 + 0.631i)6-s + (−0.235 − 0.971i)7-s + 0.353i·8-s + (−0.993 + 0.111i)9-s + 0.115i·10-s + (0.174 + 0.100i)11-s + (−0.418 + 0.273i)12-s + (0.240 − 0.138i)13-s + (0.199 − 0.678i)14-s + (−0.137 + 0.0897i)15-s + (−0.125 + 0.216i)16-s + 1.21·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$126$$    =    $$2 \cdot 3^{2} \cdot 7$$ Sign: $0.341 - 0.940i$ Analytic conductor: $$1.00611$$ Root analytic conductor: $$1.00305$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{126} (83, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 126,\ (\ :1/2),\ 0.341 - 0.940i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.17990 + 0.826990i$$ $$L(\frac12)$$ $$\approx$$ $$1.17990 + 0.826990i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.866 - 0.5i)T$$
3 $$1 + (-0.0967 - 1.72i)T$$
7 $$1 + (0.624 + 2.57i)T$$
good5 $$1 + (-0.183 - 0.317i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-0.579 - 0.334i)T + (5.5 + 9.52i)T^{2}$$
13 $$1 + (-0.867 + 0.500i)T + (6.5 - 11.2i)T^{2}$$
17 $$1 - 4.98T + 17T^{2}$$
19 $$1 + 6.35iT - 19T^{2}$$
23 $$1 + (6.66 - 3.84i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + (-1.58 - 0.914i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (5.47 - 3.16i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 + 5.16T + 37T^{2}$$
41 $$1 + (-2.15 - 3.73i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-2.24 + 3.89i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (4.16 - 7.21i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + (4.36 + 7.55i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-4.29 - 2.47i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (-5.44 - 9.43i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 5.49iT - 71T^{2}$$
73 $$1 - 4.07iT - 73T^{2}$$
79 $$1 + (4.17 - 7.23i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (8.50 - 14.7i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 + 10.7T + 89T^{2}$$
97 $$1 + (-14.9 - 8.60i)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

−13.98016901659877725078825472805, −12.69554100934106072271414950338, −11.44285607983500434070605102121, −10.48771123804329421476108444608, −9.547738620388256050669934820749, −8.164521258763114127287451048423, −6.89683945830747836106050108902, −5.55295138246210281316659527495, −4.31208636071578698735969313392, −3.21838208193317123931231656561, 1.88207767779536472002557961304, 3.45324970481293391234563155593, 5.52050253705868402202212364498, 6.23570489643954552346397555555, 7.73001675709558753051868500686, 8.861071252947852406414693415573, 10.17520593567578556389003889564, 11.62976331938711579907735386998, 12.28003552895765187908726871399, 12.93849074476169513012394296982