# Properties

 Degree 2 Conductor $2^{4} \cdot 503$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.84·3-s − 2.72·5-s + 3.71·7-s + 5.09·9-s + 3.83·11-s + 2.85·13-s + 7.74·15-s + 8.15·17-s − 3.41·19-s − 10.5·21-s − 0.915·23-s + 2.40·25-s − 5.96·27-s − 9.16·29-s − 5.69·31-s − 10.9·33-s − 10.1·35-s − 8.85·37-s − 8.11·39-s − 10.6·41-s + 11.1·43-s − 13.8·45-s + 3.44·47-s + 6.81·49-s − 23.2·51-s − 4.95·53-s − 10.4·55-s + ⋯
 L(s)  = 1 − 1.64·3-s − 1.21·5-s + 1.40·7-s + 1.69·9-s + 1.15·11-s + 0.790·13-s + 1.99·15-s + 1.97·17-s − 0.783·19-s − 2.30·21-s − 0.190·23-s + 0.480·25-s − 1.14·27-s − 1.70·29-s − 1.02·31-s − 1.90·33-s − 1.70·35-s − 1.45·37-s − 1.29·39-s − 1.66·41-s + 1.70·43-s − 2.06·45-s + 0.502·47-s + 0.973·49-s − 3.24·51-s − 0.680·53-s − 1.40·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8048$$    =    $$2^{4} \cdot 503$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{8048} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 8048,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;503\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
503 $$1 + T$$
good3 $$1 + 2.84T + 3T^{2}$$
5 $$1 + 2.72T + 5T^{2}$$
7 $$1 - 3.71T + 7T^{2}$$
11 $$1 - 3.83T + 11T^{2}$$
13 $$1 - 2.85T + 13T^{2}$$
17 $$1 - 8.15T + 17T^{2}$$
19 $$1 + 3.41T + 19T^{2}$$
23 $$1 + 0.915T + 23T^{2}$$
29 $$1 + 9.16T + 29T^{2}$$
31 $$1 + 5.69T + 31T^{2}$$
37 $$1 + 8.85T + 37T^{2}$$
41 $$1 + 10.6T + 41T^{2}$$
43 $$1 - 11.1T + 43T^{2}$$
47 $$1 - 3.44T + 47T^{2}$$
53 $$1 + 4.95T + 53T^{2}$$
59 $$1 + 2.81T + 59T^{2}$$
61 $$1 + 3.59T + 61T^{2}$$
67 $$1 - 9.01T + 67T^{2}$$
71 $$1 - 7.15T + 71T^{2}$$
73 $$1 + 7.82T + 73T^{2}$$
79 $$1 + 13.4T + 79T^{2}$$
83 $$1 - 13.5T + 83T^{2}$$
89 $$1 - 6.47T + 89T^{2}$$
97 $$1 - 16.0T + 97T^{2}$$
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\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−7.46603568532477633752989114507, −6.81316368566855053356494598013, −5.92548965263457136085193900077, −5.43079686590185439650330899877, −4.78406032447302733496722165849, −3.83674891249201893422897380886, −3.70681285271605684468007757782, −1.69767091313317834110523203051, −1.16455713802300826113658606586, 0, 1.16455713802300826113658606586, 1.69767091313317834110523203051, 3.70681285271605684468007757782, 3.83674891249201893422897380886, 4.78406032447302733496722165849, 5.43079686590185439650330899877, 5.92548965263457136085193900077, 6.81316368566855053356494598013, 7.46603568532477633752989114507