# Properties

 Degree $2$ Conductor $1050$ Sign $-0.192 - 0.981i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + 0.999·6-s + (−1.73 − 2i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.866 − 0.499i)12-s + i·13-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + (−1.5 + 2.59i)19-s + (2.5 + 0.866i)21-s − 0.999i·22-s + ⋯
 L(s)  = 1 + (−0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + 0.408·6-s + (−0.654 − 0.755i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.150 + 0.261i)11-s + (−0.249 − 0.144i)12-s + 0.277i·13-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.204 + 0.117i)18-s + (−0.344 + 0.596i)19-s + (0.545 + 0.188i)21-s − 0.213i·22-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1050$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 7$$ Sign: $-0.192 - 0.981i$ Motivic weight: $$1$$ Character: $\chi_{1050} (499, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1050,\ (\ :1/2),\ -0.192 - 0.981i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4347871061$$ $$L(\frac12)$$ $$\approx$$ $$0.4347871061$$ $$L(\frac{3}{2})$$ 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 + (0.866 - 0.5i)T$$
5 $$1$$
7 $$1 + (1.73 + 2i)T$$
good11 $$1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 - iT - 13T^{2}$$
17 $$1 + (8.5 - 14.7i)T^{2}$$
19 $$1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (6.06 + 3.5i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 - 8T + 29T^{2}$$
31 $$1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + (-9.52 - 5.5i)T + (18.5 + 32.0i)T^{2}$$
41 $$1 + 11T + 41T^{2}$$
43 $$1 - 8iT - 43T^{2}$$
47 $$1 + (4.33 + 2.5i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 + (9.52 - 5.5i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (33.5 - 58.0i)T^{2}$$
71 $$1 + 6T + 71T^{2}$$
73 $$1 + (5.19 - 3i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 8iT - 83T^{2}$$
89 $$1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 - 16iT - 97T^{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

−10.04007331385252304572081229304, −9.711896020228065602674908338159, −8.520167814969533306583362223686, −7.79540939934468780821060490540, −6.61626681642642048571109342739, −6.27247098058301995790817486838, −4.71165555941424503048609533944, −3.96160036411659492744353436473, −2.81298008647284192148154497594, −1.26861404259426359437036417380, 0.27335889485403576172885470641, 1.93264561848697945462907098021, 3.15955263528094768091559034885, 4.63722305760108412589690753785, 5.75560105492558559235404173295, 6.25656280712652071989658912377, 7.10673217437251471965409218023, 8.104171908779820647803368011115, 8.772509503233306378477662116910, 9.730032836443808680045021894207