# Properties

 Label 2-103-1.1-c1-0-2 Degree $2$ Conductor $103$ Sign $-1$ Analytic cond. $0.822459$ Root an. cond. $0.906895$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.61·2-s − 3-s + 4.85·4-s − 0.381·5-s + 2.61·6-s − 7-s − 7.47·8-s − 2·9-s + 10-s − 2.61·11-s − 4.85·12-s − 4.85·13-s + 2.61·14-s + 0.381·15-s + 9.85·16-s − 5.61·17-s + 5.23·18-s + 5.85·19-s − 1.85·20-s + 21-s + 6.85·22-s + 4.47·23-s + 7.47·24-s − 4.85·25-s + 12.7·26-s + 5·27-s − 4.85·28-s + ⋯
 L(s)  = 1 − 1.85·2-s − 0.577·3-s + 2.42·4-s − 0.170·5-s + 1.06·6-s − 0.377·7-s − 2.64·8-s − 0.666·9-s + 0.316·10-s − 0.789·11-s − 1.40·12-s − 1.34·13-s + 0.699·14-s + 0.0986·15-s + 2.46·16-s − 1.36·17-s + 1.23·18-s + 1.34·19-s − 0.414·20-s + 0.218·21-s + 1.46·22-s + 0.932·23-s + 1.52·24-s − 0.970·25-s + 2.49·26-s + 0.962·27-s − 0.917·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$103$$ Sign: $-1$ Analytic conductor: $$0.822459$$ Root analytic conductor: $$0.906895$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 103,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad103 $$1 + T$$
good2 $$1 + 2.61T + 2T^{2}$$
3 $$1 + T + 3T^{2}$$
5 $$1 + 0.381T + 5T^{2}$$
7 $$1 + T + 7T^{2}$$
11 $$1 + 2.61T + 11T^{2}$$
13 $$1 + 4.85T + 13T^{2}$$
17 $$1 + 5.61T + 17T^{2}$$
19 $$1 - 5.85T + 19T^{2}$$
23 $$1 - 4.47T + 23T^{2}$$
29 $$1 + 5.23T + 29T^{2}$$
31 $$1 + 6.70T + 31T^{2}$$
37 $$1 - 6.70T + 37T^{2}$$
41 $$1 - 8.94T + 41T^{2}$$
43 $$1 + 8.70T + 43T^{2}$$
47 $$1 - 4.09T + 47T^{2}$$
53 $$1 - 1.09T + 53T^{2}$$
59 $$1 - 6.38T + 59T^{2}$$
61 $$1 - 4.14T + 61T^{2}$$
67 $$1 - 14.4T + 67T^{2}$$
71 $$1 + 4.09T + 71T^{2}$$
73 $$1 + 10.8T + 73T^{2}$$
79 $$1 + 6.56T + 79T^{2}$$
83 $$1 + 6.32T + 83T^{2}$$
89 $$1 + 2.29T + 89T^{2}$$
97 $$1 + 1.70T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$