Properties

 Label 2-1815-1.1-c1-0-12 Degree $2$ Conductor $1815$ Sign $1$ Analytic cond. $14.4928$ Root an. cond. $3.80694$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 0.414·2-s − 3-s − 1.82·4-s − 5-s + 0.414·6-s + 4.82·7-s + 1.58·8-s + 9-s + 0.414·10-s + 1.82·12-s − 5.65·13-s − 1.99·14-s + 15-s + 3·16-s + 6.82·17-s − 0.414·18-s + 1.17·19-s + 1.82·20-s − 4.82·21-s − 4·23-s − 1.58·24-s + 25-s + 2.34·26-s − 27-s − 8.82·28-s − 0.828·29-s − 0.414·30-s + ⋯
 L(s)  = 1 − 0.292·2-s − 0.577·3-s − 0.914·4-s − 0.447·5-s + 0.169·6-s + 1.82·7-s + 0.560·8-s + 0.333·9-s + 0.130·10-s + 0.527·12-s − 1.56·13-s − 0.534·14-s + 0.258·15-s + 0.750·16-s + 1.65·17-s − 0.0976·18-s + 0.268·19-s + 0.408·20-s − 1.05·21-s − 0.834·23-s − 0.323·24-s + 0.200·25-s + 0.459·26-s − 0.192·27-s − 1.66·28-s − 0.153·29-s − 0.0756·30-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1815$$    =    $$3 \cdot 5 \cdot 11^{2}$$ Sign: $1$ Analytic conductor: $$14.4928$$ Root analytic conductor: $$3.80694$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1815} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 1815,\ (\ :1/2),\ 1)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.008641514$$ $$L(\frac12)$$ $$\approx$$ $$1.008641514$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + T$$
5 $$1 + T$$
11 $$1$$
good2 $$1 + 0.414T + 2T^{2}$$
7 $$1 - 4.82T + 7T^{2}$$
13 $$1 + 5.65T + 13T^{2}$$
17 $$1 - 6.82T + 17T^{2}$$
19 $$1 - 1.17T + 19T^{2}$$
23 $$1 + 4T + 23T^{2}$$
29 $$1 + 0.828T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 0.343T + 37T^{2}$$
41 $$1 - 0.828T + 41T^{2}$$
43 $$1 - 3.17T + 43T^{2}$$
47 $$1 + 4T + 47T^{2}$$
53 $$1 + 13.3T + 53T^{2}$$
59 $$1 + 4T + 59T^{2}$$
61 $$1 - 0.343T + 61T^{2}$$
67 $$1 - 5.65T + 67T^{2}$$
71 $$1 - 13.6T + 71T^{2}$$
73 $$1 - 11.3T + 73T^{2}$$
79 $$1 - 8.48T + 79T^{2}$$
83 $$1 - 10T + 83T^{2}$$
89 $$1 + 7.65T + 89T^{2}$$
97 $$1 - 0.343T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$