# Properties

 Label 2-325-13.3-c1-0-13 Degree $2$ Conductor $325$ Sign $0.859 + 0.511i$ Analytic cond. $2.59513$ Root an. cond. $1.61094$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.809 + 1.40i)2-s + (−1.11 − 1.93i)3-s + (−0.309 + 0.535i)4-s + (1.80 − 3.13i)6-s + (−0.118 + 0.204i)7-s + 2.23·8-s + (−1 + 1.73i)9-s + (−2.11 − 3.66i)11-s + 1.38·12-s + (1 − 3.46i)13-s − 0.381·14-s + (2.42 + 4.20i)16-s + (2.73 − 4.73i)17-s − 3.23·18-s + (0.118 − 0.204i)19-s + ⋯
 L(s)  = 1 + (0.572 + 0.990i)2-s + (−0.645 − 1.11i)3-s + (−0.154 + 0.267i)4-s + (0.738 − 1.27i)6-s + (−0.0446 + 0.0772i)7-s + 0.790·8-s + (−0.333 + 0.577i)9-s + (−0.638 − 1.10i)11-s + 0.398·12-s + (0.277 − 0.960i)13-s − 0.102·14-s + (0.606 + 1.05i)16-s + (0.663 − 1.14i)17-s − 0.762·18-s + (0.0270 − 0.0469i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$325$$    =    $$5^{2} \cdot 13$$ Sign: $0.859 + 0.511i$ Analytic conductor: $$2.59513$$ Root analytic conductor: $$1.61094$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{325} (276, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 325,\ (\ :1/2),\ 0.859 + 0.511i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.44859 - 0.398123i$$ $$L(\frac12)$$ $$\approx$$ $$1.44859 - 0.398123i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
13 $$1 + (-1 + 3.46i)T$$
good2 $$1 + (-0.809 - 1.40i)T + (-1 + 1.73i)T^{2}$$
3 $$1 + (1.11 + 1.93i)T + (-1.5 + 2.59i)T^{2}$$
7 $$1 + (0.118 - 0.204i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + (2.11 + 3.66i)T + (-5.5 + 9.52i)T^{2}$$
17 $$1 + (-2.73 + 4.73i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-0.118 + 0.204i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-4.11 - 7.13i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (0.736 + 1.27i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (-2.97 - 5.14i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (0.881 - 1.52i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + 12.9T + 47T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 + (6.35 - 11.0i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-6.20 + 10.7i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (-5.35 - 9.27i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (-0.881 + 1.52i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 - 6T + 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 8.94T + 83T^{2}$$
89 $$1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (-2.73 + 4.73i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$