# Properties

 Degree 2 Conductor $2^{2} \cdot 17 \cdot 59$ Sign $-0.699 + 0.714i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.593i·3-s − 3.87i·5-s − 2.92i·7-s + 2.64·9-s + 4.95i·11-s − 3.80·13-s + 2.29·15-s + (2.88 − 2.94i)17-s + 1.82·19-s + 1.73·21-s − 8.40i·23-s − 10.0·25-s + 3.35i·27-s + 1.64i·29-s − 7.62i·31-s + ⋯
 L(s)  = 1 + 0.342i·3-s − 1.73i·5-s − 1.10i·7-s + 0.882·9-s + 1.49i·11-s − 1.05·13-s + 0.593·15-s + (0.699 − 0.714i)17-s + 0.417·19-s + 0.379·21-s − 1.75i·23-s − 2.00·25-s + 0.644i·27-s + 0.305i·29-s − 1.36i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4012$$    =    $$2^{2} \cdot 17 \cdot 59$$ $$\varepsilon$$ = $-0.699 + 0.714i$ motivic weight = $$1$$ character : $\chi_{4012} (237, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4012,\ (\ :1/2),\ -0.699 + 0.714i)$ $L(1)$ $\approx$ $1.611405055$ $L(\frac12)$ $\approx$ $1.611405055$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;17,\;59\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
17 $$1 + (-2.88 + 2.94i)T$$
59 $$1 - T$$
good3 $$1 - 0.593iT - 3T^{2}$$
5 $$1 + 3.87iT - 5T^{2}$$
7 $$1 + 2.92iT - 7T^{2}$$
11 $$1 - 4.95iT - 11T^{2}$$
13 $$1 + 3.80T + 13T^{2}$$
19 $$1 - 1.82T + 19T^{2}$$
23 $$1 + 8.40iT - 23T^{2}$$
29 $$1 - 1.64iT - 29T^{2}$$
31 $$1 + 7.62iT - 31T^{2}$$
37 $$1 + 0.0884iT - 37T^{2}$$
41 $$1 + 2.48iT - 41T^{2}$$
43 $$1 - 10.2T + 43T^{2}$$
47 $$1 + 1.02T + 47T^{2}$$
53 $$1 - 0.471T + 53T^{2}$$
61 $$1 - 6.85iT - 61T^{2}$$
67 $$1 + 12.9T + 67T^{2}$$
71 $$1 + 8.36iT - 71T^{2}$$
73 $$1 - 7.15iT - 73T^{2}$$
79 $$1 + 8.40iT - 79T^{2}$$
83 $$1 - 2.81T + 83T^{2}$$
89 $$1 + 12.6T + 89T^{2}$$
97 $$1 + 2.93iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}