Properties

 Label 2-15e2-15.2-c1-0-0 Degree $2$ Conductor $225$ Sign $-0.876 - 0.481i$ Analytic cond. $1.79663$ Root an. cond. $1.34038$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−1.73 + 1.73i)2-s − 3.99i·4-s + (1.22 + 1.22i)7-s + (3.46 + 3.46i)8-s + 4.24i·11-s + (−3.67 + 3.67i)13-s − 4.24·14-s − 3.99·16-s + (1.73 − 1.73i)17-s + 5i·19-s + (−7.34 − 7.34i)22-s + (−1.73 − 1.73i)23-s − 12.7i·26-s + (4.89 − 4.89i)28-s − 4.24·29-s + ⋯
 L(s)  = 1 + (−1.22 + 1.22i)2-s − 1.99i·4-s + (0.462 + 0.462i)7-s + (1.22 + 1.22i)8-s + 1.27i·11-s + (−1.01 + 1.01i)13-s − 1.13·14-s − 0.999·16-s + (0.420 − 0.420i)17-s + 1.14i·19-s + (−1.56 − 1.56i)22-s + (−0.361 − 0.361i)23-s − 2.49i·26-s + (0.925 − 0.925i)28-s − 0.787·29-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $-0.876 - 0.481i$ Analytic conductor: $$1.79663$$ Root analytic conductor: $$1.34038$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{225} (107, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 225,\ (\ :1/2),\ -0.876 - 0.481i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$0.142229 + 0.554215i$$ $$L(\frac12)$$ $$\approx$$ $$0.142229 + 0.554215i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1 + (1.73 - 1.73i)T - 2iT^{2}$$
7 $$1 + (-1.22 - 1.22i)T + 7iT^{2}$$
11 $$1 - 4.24iT - 11T^{2}$$
13 $$1 + (3.67 - 3.67i)T - 13iT^{2}$$
17 $$1 + (-1.73 + 1.73i)T - 17iT^{2}$$
19 $$1 - 5iT - 19T^{2}$$
23 $$1 + (1.73 + 1.73i)T + 23iT^{2}$$
29 $$1 + 4.24T + 29T^{2}$$
31 $$1 - T + 31T^{2}$$
37 $$1 + (-2.44 - 2.44i)T + 37iT^{2}$$
41 $$1 - 8.48iT - 41T^{2}$$
43 $$1 + (1.22 - 1.22i)T - 43iT^{2}$$
47 $$1 + (-5.19 + 5.19i)T - 47iT^{2}$$
53 $$1 + (-6.92 - 6.92i)T + 53iT^{2}$$
59 $$1 - 12.7T + 59T^{2}$$
61 $$1 + 7T + 61T^{2}$$
67 $$1 + (-3.67 - 3.67i)T + 67iT^{2}$$
71 $$1 + 8.48iT - 71T^{2}$$
73 $$1 + (-2.44 + 2.44i)T - 73iT^{2}$$
79 $$1 + 2iT - 79T^{2}$$
83 $$1 + (1.73 + 1.73i)T + 83iT^{2}$$
89 $$1 + 8.48T + 89T^{2}$$
97 $$1 + (8.57 + 8.57i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$