# Properties

 Label 2-14e2-7.4-c5-0-14 Degree $2$ Conductor $196$ Sign $-0.968 - 0.250i$ Analytic cond. $31.4352$ Root an. cond. $5.60671$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + 1.73i)3-s + (−48 − 83.1i)5-s + (119.5 + 206. i)9-s + (360 − 623. i)11-s − 572·13-s + 192·15-s + (627 − 1.08e3i)17-s + (−47 − 81.4i)19-s + (−48 − 83.1i)23-s + (−3.04e3 + 5.27e3i)25-s − 964·27-s − 4.37e3·29-s + (−3.12e3 + 5.40e3i)31-s + (720 + 1.24e3i)33-s + (5.39e3 + 9.35e3i)37-s + ⋯
 L(s)  = 1 + (−0.0641 + 0.111i)3-s + (−0.858 − 1.48i)5-s + (0.491 + 0.851i)9-s + (0.897 − 1.55i)11-s − 0.938·13-s + 0.220·15-s + (0.526 − 0.911i)17-s + (−0.0298 − 0.0517i)19-s + (−0.0189 − 0.0327i)23-s + (−0.974 + 1.68i)25-s − 0.254·27-s − 0.965·29-s + (−0.583 + 1.01i)31-s + (0.115 + 0.199i)33-s + (0.648 + 1.12i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$196$$    =    $$2^{2} \cdot 7^{2}$$ Sign: $-0.968 - 0.250i$ Analytic conductor: $$31.4352$$ Root analytic conductor: $$5.60671$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{196} (165, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 196,\ (\ :5/2),\ -0.968 - 0.250i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.4981856279$$ $$L(\frac12)$$ $$\approx$$ $$0.4981856279$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1$$
good3 $$1 + (1 - 1.73i)T + (-121.5 - 210. i)T^{2}$$
5 $$1 + (48 + 83.1i)T + (-1.56e3 + 2.70e3i)T^{2}$$
11 $$1 + (-360 + 623. i)T + (-8.05e4 - 1.39e5i)T^{2}$$
13 $$1 + 572T + 3.71e5T^{2}$$
17 $$1 + (-627 + 1.08e3i)T + (-7.09e5 - 1.22e6i)T^{2}$$
19 $$1 + (47 + 81.4i)T + (-1.23e6 + 2.14e6i)T^{2}$$
23 $$1 + (48 + 83.1i)T + (-3.21e6 + 5.57e6i)T^{2}$$
29 $$1 + 4.37e3T + 2.05e7T^{2}$$
31 $$1 + (3.12e3 - 5.40e3i)T + (-1.43e7 - 2.47e7i)T^{2}$$
37 $$1 + (-5.39e3 - 9.35e3i)T + (-3.46e7 + 6.00e7i)T^{2}$$
41 $$1 + 1.20e4T + 1.15e8T^{2}$$
43 $$1 + 9.16e3T + 1.47e8T^{2}$$
47 $$1 + (1.29e4 + 2.23e4i)T + (-1.14e8 + 1.98e8i)T^{2}$$
53 $$1 + (507 - 878. i)T + (-2.09e8 - 3.62e8i)T^{2}$$
59 $$1 + (-621 + 1.07e3i)T + (-3.57e8 - 6.19e8i)T^{2}$$
61 $$1 + (-3.79e3 - 6.57e3i)T + (-4.22e8 + 7.31e8i)T^{2}$$
67 $$1 + (2.05e4 - 3.56e4i)T + (-6.75e8 - 1.16e9i)T^{2}$$
71 $$1 + 3.76e4T + 1.80e9T^{2}$$
73 $$1 + (6.71e3 - 1.16e4i)T + (-1.03e9 - 1.79e9i)T^{2}$$
79 $$1 + (3.12e3 + 5.41e3i)T + (-1.53e9 + 2.66e9i)T^{2}$$
83 $$1 - 2.52e4T + 3.93e9T^{2}$$
89 $$1 + (2.25e4 + 3.90e4i)T + (-2.79e9 + 4.83e9i)T^{2}$$
97 $$1 + 1.07e5T + 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$