# Properties

 Degree 2 Conductor 2 Sign $1$ Motivic weight 41 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.04e6·2-s + 1.05e10·3-s + 1.09e12·4-s − 2.54e14·5-s + 1.10e16·6-s + 2.96e17·7-s + 1.15e18·8-s + 7.54e19·9-s − 2.66e20·10-s − 1.09e21·11-s + 1.16e22·12-s + 8.09e22·13-s + 3.10e23·14-s − 2.69e24·15-s + 1.20e24·16-s − 5.33e24·17-s + 7.91e25·18-s − 3.50e25·19-s − 2.79e26·20-s + 3.13e27·21-s − 1.14e27·22-s + 6.68e27·23-s + 1.21e28·24-s + 1.91e28·25-s + 8.48e28·26-s + 4.12e29·27-s + 3.25e29·28-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1.75·3-s + 0.5·4-s − 1.19·5-s + 1.23·6-s + 1.40·7-s + 0.353·8-s + 2.06·9-s − 0.843·10-s − 0.491·11-s + 0.876·12-s + 1.18·13-s + 0.991·14-s − 2.08·15-s + 0.250·16-s − 0.318·17-s + 1.46·18-s − 0.214·19-s − 0.596·20-s + 2.45·21-s − 0.347·22-s + 0.812·23-s + 0.619·24-s + 0.421·25-s + 0.835·26-s + 1.87·27-s + 0.701·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2$$ $$\varepsilon$$ = $1$ motivic weight = $$41$$ character : $\chi_{2} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 2,\ (\ :41/2),\ 1)$ $L(21)$ $\approx$ $5.07957$ $L(\frac12)$ $\approx$ $5.07957$ $L(\frac{43}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 2$, $$F_p$$ is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 - 1.04e6T$$
good3 $$1 - 1.05e10T + 3.64e19T^{2}$$
5 $$1 + 2.54e14T + 4.54e28T^{2}$$
7 $$1 - 2.96e17T + 4.45e34T^{2}$$
11 $$1 + 1.09e21T + 4.97e42T^{2}$$
13 $$1 - 8.09e22T + 4.69e45T^{2}$$
17 $$1 + 5.33e24T + 2.80e50T^{2}$$
19 $$1 + 3.50e25T + 2.68e52T^{2}$$
23 $$1 - 6.68e27T + 6.77e55T^{2}$$
29 $$1 + 1.63e30T + 9.08e59T^{2}$$
31 $$1 - 2.77e30T + 1.39e61T^{2}$$
37 $$1 - 4.65e31T + 1.97e64T^{2}$$
41 $$1 + 8.83e32T + 1.33e66T^{2}$$
43 $$1 + 3.69e32T + 9.38e66T^{2}$$
47 $$1 + 1.89e34T + 3.59e68T^{2}$$
53 $$1 + 2.01e35T + 4.95e70T^{2}$$
59 $$1 - 5.07e34T + 4.02e72T^{2}$$
61 $$1 + 6.57e36T + 1.57e73T^{2}$$
67 $$1 - 5.28e37T + 7.39e74T^{2}$$
71 $$1 + 1.04e38T + 7.97e75T^{2}$$
73 $$1 + 2.83e38T + 2.49e76T^{2}$$
79 $$1 + 6.59e38T + 6.34e77T^{2}$$
83 $$1 - 1.30e39T + 4.81e78T^{2}$$
89 $$1 - 2.16e39T + 8.41e79T^{2}$$
97 $$1 + 8.40e39T + 2.86e81T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}