# Properties

 Label 2-98-7.4-c1-0-2 Degree $2$ Conductor $98$ Sign $0.991 - 0.126i$ Analytic cond. $0.782533$ Root an. cond. $0.884609$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 + 0.866i)2-s + (1 − 1.73i)3-s + (−0.499 + 0.866i)4-s + 1.99·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.999 + 1.73i)12-s − 4·13-s + (−0.5 − 0.866i)16-s + (−3 + 5.19i)17-s + (0.499 − 0.866i)18-s + (−1 − 1.73i)19-s + (−0.999 + 1.73i)24-s + (2.5 − 4.33i)25-s + (−2 − 3.46i)26-s + 4.00·27-s + ⋯
 L(s)  = 1 + (0.353 + 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 + 0.433i)4-s + 0.816·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.288 + 0.499i)12-s − 1.10·13-s + (−0.125 − 0.216i)16-s + (−0.727 + 1.26i)17-s + (0.117 − 0.204i)18-s + (−0.229 − 0.397i)19-s + (−0.204 + 0.353i)24-s + (0.5 − 0.866i)25-s + (−0.392 − 0.679i)26-s + 0.769·27-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$98$$    =    $$2 \cdot 7^{2}$$ Sign: $0.991 - 0.126i$ Analytic conductor: $$0.782533$$ Root analytic conductor: $$0.884609$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{98} (67, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 98,\ (\ :1/2),\ 0.991 - 0.126i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.29448 + 0.0821484i$$ $$L(\frac12)$$ $$\approx$$ $$1.29448 + 0.0821484i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.5 - 0.866i)T$$
7 $$1$$
good3 $$1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-5.5 - 9.52i)T^{2}$$
13 $$1 + 4T + 13T^{2}$$
17 $$1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-11.5 + 19.9i)T^{2}$$
29 $$1 + 6T + 29T^{2}$$
31 $$1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 6T + 41T^{2}$$
43 $$1 - 8T + 43T^{2}$$
47 $$1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 6T + 83T^{2}$$
89 $$1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + 10T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$