# Properties

 Label 2-52-52.7-c1-0-0 Degree $2$ Conductor $52$ Sign $0.884 - 0.466i$ Analytic cond. $0.415222$ Root an. cond. $0.644377$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.36 + 0.366i)2-s + (1.73 − i)4-s + (2.36 + 2.36i)5-s + (−1.99 + 2i)8-s + (−1.5 − 2.59i)9-s + (−4.09 − 2.36i)10-s + (−1.59 − 3.23i)13-s + (1.99 − 3.46i)16-s + (−5.13 + 2.96i)17-s + (3 + 3i)18-s + (6.46 + 1.73i)20-s + 6.19i·25-s + (3.36 + 3.83i)26-s + (5.33 − 9.23i)29-s + (−1.46 + 5.46i)32-s + ⋯
 L(s)  = 1 + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)4-s + (1.05 + 1.05i)5-s + (−0.707 + 0.707i)8-s + (−0.5 − 0.866i)9-s + (−1.29 − 0.748i)10-s + (−0.443 − 0.896i)13-s + (0.499 − 0.866i)16-s + (−1.24 + 0.718i)17-s + (0.707 + 0.707i)18-s + (1.44 + 0.387i)20-s + 1.23i·25-s + (0.660 + 0.751i)26-s + (0.989 − 1.71i)29-s + (−0.258 + 0.965i)32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$52$$    =    $$2^{2} \cdot 13$$ Sign: $0.884 - 0.466i$ Analytic conductor: $$0.415222$$ Root analytic conductor: $$0.644377$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{52} (7, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 52,\ (\ :1/2),\ 0.884 - 0.466i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.613181 + 0.151676i$$ $$L(\frac12)$$ $$\approx$$ $$0.613181 + 0.151676i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.36 - 0.366i)T$$
13 $$1 + (1.59 + 3.23i)T$$
good3 $$1 + (1.5 + 2.59i)T^{2}$$
5 $$1 + (-2.36 - 2.36i)T + 5iT^{2}$$
7 $$1 + (-6.06 - 3.5i)T^{2}$$
11 $$1 + (9.52 - 5.5i)T^{2}$$
17 $$1 + (5.13 - 2.96i)T + (8.5 - 14.7i)T^{2}$$
19 $$1 + (16.4 + 9.5i)T^{2}$$
23 $$1 + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (-5.33 + 9.23i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 - 31iT^{2}$$
37 $$1 + (-1.30 - 4.86i)T + (-32.0 + 18.5i)T^{2}$$
41 $$1 + (11.3 - 3.03i)T + (35.5 - 20.5i)T^{2}$$
43 $$1 + (-21.5 + 37.2i)T^{2}$$
47 $$1 + 47iT^{2}$$
53 $$1 - 3.53T + 53T^{2}$$
59 $$1 + (-51.0 - 29.5i)T^{2}$$
61 $$1 + (-7.69 - 13.3i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-58.0 + 33.5i)T^{2}$$
71 $$1 + (61.4 + 35.5i)T^{2}$$
73 $$1 + (-1.16 + 1.16i)T - 73iT^{2}$$
79 $$1 - 79T^{2}$$
83 $$1 - 83iT^{2}$$
89 $$1 + (-1.09 - 4.09i)T + (-77.0 + 44.5i)T^{2}$$
97 $$1 + (-1.83 + 6.83i)T + (-84.0 - 48.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$