# Properties

 Degree $2$ Conductor $1$ Sign $-1$ Motivic weight $105$ Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.72e15·2-s − 1.28e25·3-s − 3.75e31·4-s − 7.91e36·5-s − 2.21e40·6-s + 2.76e44·7-s − 1.34e47·8-s + 4.01e49·9-s − 1.36e52·10-s − 6.07e54·11-s + 4.83e56·12-s + 2.29e58·13-s + 4.76e59·14-s + 1.01e62·15-s + 1.29e63·16-s − 1.76e64·17-s + 6.92e64·18-s + 1.67e67·19-s + 2.97e68·20-s − 3.55e69·21-s − 1.04e70·22-s + 4.70e71·23-s + 1.73e72·24-s + 3.79e73·25-s + 3.95e73·26-s + 1.09e75·27-s − 1.03e76·28-s + ⋯
 L(s)  = 1 + 0.270·2-s − 1.14·3-s − 0.926·4-s − 1.59·5-s − 0.311·6-s + 1.18·7-s − 0.521·8-s + 0.320·9-s − 0.431·10-s − 1.28·11-s + 1.06·12-s + 0.756·13-s + 0.320·14-s + 1.83·15-s + 0.785·16-s − 0.445·17-s + 0.0868·18-s + 1.22·19-s + 1.47·20-s − 1.36·21-s − 0.348·22-s + 1.52·23-s + 0.599·24-s + 1.54·25-s + 0.204·26-s + 0.780·27-s − 1.09·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(106-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+52.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1$$ Sign: $-1$ Motivic weight: $$105$$ Character: $\chi_{1} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1,\ (\ :105/2),\ -1)$$

## Particular Values

 $$L(53)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{107}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
good2 $$1 - 1.72e15T + 4.05e31T^{2}$$
3 $$1 + 1.28e25T + 1.25e50T^{2}$$
5 $$1 + 7.91e36T + 2.46e73T^{2}$$
7 $$1 - 2.76e44T + 5.43e88T^{2}$$
11 $$1 + 6.07e54T + 2.21e109T^{2}$$
13 $$1 - 2.29e58T + 9.20e116T^{2}$$
17 $$1 + 1.76e64T + 1.57e129T^{2}$$
19 $$1 - 1.67e67T + 1.85e134T^{2}$$
23 $$1 - 4.70e71T + 9.58e142T^{2}$$
29 $$1 + 2.66e76T + 3.56e153T^{2}$$
31 $$1 + 3.05e78T + 3.91e156T^{2}$$
37 $$1 - 8.35e81T + 4.58e164T^{2}$$
41 $$1 + 4.13e84T + 2.19e169T^{2}$$
43 $$1 - 2.16e85T + 3.26e171T^{2}$$
47 $$1 + 2.62e87T + 3.71e175T^{2}$$
53 $$1 + 2.71e90T + 1.11e181T^{2}$$
59 $$1 - 1.20e93T + 8.69e185T^{2}$$
61 $$1 - 7.31e93T + 2.88e187T^{2}$$
67 $$1 + 2.41e95T + 5.46e191T^{2}$$
71 $$1 + 9.07e96T + 2.41e194T^{2}$$
73 $$1 + 9.30e97T + 4.45e195T^{2}$$
79 $$1 - 5.92e99T + 1.78e199T^{2}$$
83 $$1 + 1.53e99T + 3.18e201T^{2}$$
89 $$1 - 2.87e102T + 4.85e204T^{2}$$
97 $$1 - 8.84e103T + 4.08e208T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$