# Properties

 Label 2-136-17.2-c1-0-0 Degree $2$ Conductor $136$ Sign $-0.197 - 0.980i$ Analytic cond. $1.08596$ Root an. cond. $1.04209$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.70 + 0.707i)3-s + (0.292 + 0.707i)5-s + (−1.70 + 4.12i)7-s + (0.292 − 0.292i)9-s + (1.70 + 0.707i)11-s + (−1 − 0.999i)15-s + (1 + 4i)17-s + (−4.41 − 4.41i)19-s − 8.24i·21-s + (4.53 + 1.87i)23-s + (3.12 − 3.12i)25-s + (1.82 − 4.41i)27-s + (−0.878 − 2.12i)29-s + (5.12 − 2.12i)31-s − 3.41·33-s + ⋯
 L(s)  = 1 + (−0.985 + 0.408i)3-s + (0.130 + 0.316i)5-s + (−0.645 + 1.55i)7-s + (0.0976 − 0.0976i)9-s + (0.514 + 0.213i)11-s + (−0.258 − 0.258i)15-s + (0.242 + 0.970i)17-s + (−1.01 − 1.01i)19-s − 1.79i·21-s + (0.945 + 0.391i)23-s + (0.624 − 0.624i)25-s + (0.351 − 0.849i)27-s + (−0.163 − 0.393i)29-s + (0.919 − 0.381i)31-s − 0.594·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$136$$    =    $$2^{3} \cdot 17$$ Sign: $-0.197 - 0.980i$ Analytic conductor: $$1.08596$$ Root analytic conductor: $$1.04209$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{136} (121, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 136,\ (\ :1/2),\ -0.197 - 0.980i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.445351 + 0.543824i$$ $$L(\frac12)$$ $$\approx$$ $$0.445351 + 0.543824i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
17 $$1 + (-1 - 4i)T$$
good3 $$1 + (1.70 - 0.707i)T + (2.12 - 2.12i)T^{2}$$
5 $$1 + (-0.292 - 0.707i)T + (-3.53 + 3.53i)T^{2}$$
7 $$1 + (1.70 - 4.12i)T + (-4.94 - 4.94i)T^{2}$$
11 $$1 + (-1.70 - 0.707i)T + (7.77 + 7.77i)T^{2}$$
13 $$1 - 13T^{2}$$
19 $$1 + (4.41 + 4.41i)T + 19iT^{2}$$
23 $$1 + (-4.53 - 1.87i)T + (16.2 + 16.2i)T^{2}$$
29 $$1 + (0.878 + 2.12i)T + (-20.5 + 20.5i)T^{2}$$
31 $$1 + (-5.12 + 2.12i)T + (21.9 - 21.9i)T^{2}$$
37 $$1 + (-1.70 + 0.707i)T + (26.1 - 26.1i)T^{2}$$
41 $$1 + (3.70 - 8.94i)T + (-28.9 - 28.9i)T^{2}$$
43 $$1 + (1.24 - 1.24i)T - 43iT^{2}$$
47 $$1 - 7.17iT - 47T^{2}$$
53 $$1 + (-7.82 - 7.82i)T + 53iT^{2}$$
59 $$1 + (-8.41 + 8.41i)T - 59iT^{2}$$
61 $$1 + (4.87 - 11.7i)T + (-43.1 - 43.1i)T^{2}$$
67 $$1 + 1.65T + 67T^{2}$$
71 $$1 + (-2.29 + 0.949i)T + (50.2 - 50.2i)T^{2}$$
73 $$1 + (5.36 + 12.9i)T + (-51.6 + 51.6i)T^{2}$$
79 $$1 + (-12.5 - 5.19i)T + (55.8 + 55.8i)T^{2}$$
83 $$1 + (-1.24 - 1.24i)T + 83iT^{2}$$
89 $$1 + 9.65iT - 89T^{2}$$
97 $$1 + (-0.778 - 1.87i)T + (-68.5 + 68.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$