Properties

 Label 2-55-55.54-c4-0-13 Degree $2$ Conductor $55$ Sign $1$ Analytic cond. $5.68534$ Root an. cond. $2.38439$ Motivic weight $4$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

Origins

Dirichlet series

 L(s)  = 1 + 3·2-s − 7·4-s + 25·5-s + 78·7-s − 69·8-s + 81·9-s + 75·10-s + 121·11-s − 162·13-s + 234·14-s − 95·16-s − 402·17-s + 243·18-s − 175·20-s + 363·22-s + 625·25-s − 486·26-s − 546·28-s − 1.59e3·31-s + 819·32-s − 1.20e3·34-s + 1.95e3·35-s − 567·36-s − 1.72e3·40-s − 3.52e3·43-s − 847·44-s + 2.02e3·45-s + ⋯
 L(s)  = 1 + 3/4·2-s − 0.437·4-s + 5-s + 1.59·7-s − 1.07·8-s + 9-s + 3/4·10-s + 11-s − 0.958·13-s + 1.19·14-s − 0.371·16-s − 1.39·17-s + 3/4·18-s − 0.437·20-s + 3/4·22-s + 25-s − 0.718·26-s − 0.696·28-s − 1.66·31-s + 0.799·32-s − 1.04·34-s + 1.59·35-s − 0.437·36-s − 1.07·40-s − 1.90·43-s − 0.437·44-s + 45-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$55$$    =    $$5 \cdot 11$$ Sign: $1$ Analytic conductor: $$5.68534$$ Root analytic conductor: $$2.38439$$ Motivic weight: $$4$$ Rational: yes Arithmetic: yes Character: $\chi_{55} (54, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 55,\ (\ :2),\ 1)$$

Particular Values

 $$L(\frac{5}{2})$$ $$\approx$$ $$2.512007194$$ $$L(\frac12)$$ $$\approx$$ $$2.512007194$$ $$L(3)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 - p^{2} T$$
11 $$1 - p^{2} T$$
good2 $$1 - 3 T + p^{4} T^{2}$$
3 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
7 $$1 - 78 T + p^{4} T^{2}$$
13 $$1 + 162 T + p^{4} T^{2}$$
17 $$1 + 402 T + p^{4} T^{2}$$
19 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
23 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
29 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
31 $$1 + 1598 T + p^{4} T^{2}$$
37 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
41 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
43 $$1 + 3522 T + p^{4} T^{2}$$
47 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
53 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
59 $$1 - 3442 T + p^{4} T^{2}$$
61 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
67 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
71 $$1 + 3998 T + p^{4} T^{2}$$
73 $$1 - 10638 T + p^{4} T^{2}$$
79 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
83 $$1 + 13602 T + p^{4} T^{2}$$
89 $$1 + 15838 T + p^{4} T^{2}$$
97 $$( 1 - p^{2} T )( 1 + p^{2} T )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$