# Properties

 Degree 2 Conductor $3 \cdot 17 \cdot 157$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.78·2-s − 3-s + 5.77·4-s − 2.00·5-s − 2.78·6-s + 0.938·7-s + 10.5·8-s + 9-s − 5.59·10-s − 5.68·11-s − 5.77·12-s + 0.884·13-s + 2.61·14-s + 2.00·15-s + 17.8·16-s − 17-s + 2.78·18-s − 2.71·19-s − 11.5·20-s − 0.938·21-s − 15.8·22-s − 5.54·23-s − 10.5·24-s − 0.977·25-s + 2.46·26-s − 27-s + 5.42·28-s + ⋯
 L(s)  = 1 + 1.97·2-s − 0.577·3-s + 2.88·4-s − 0.896·5-s − 1.13·6-s + 0.354·7-s + 3.72·8-s + 0.333·9-s − 1.76·10-s − 1.71·11-s − 1.66·12-s + 0.245·13-s + 0.699·14-s + 0.517·15-s + 4.45·16-s − 0.242·17-s + 0.657·18-s − 0.622·19-s − 2.59·20-s − 0.204·21-s − 3.38·22-s − 1.15·23-s − 2.15·24-s − 0.195·25-s + 0.483·26-s − 0.192·27-s + 1.02·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8007$$    =    $$3 \cdot 17 \cdot 157$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{8007} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 8007,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;17,\;157\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + T$$
17 $$1 + T$$
157 $$1 + T$$
good2 $$1 - 2.78T + 2T^{2}$$
5 $$1 + 2.00T + 5T^{2}$$
7 $$1 - 0.938T + 7T^{2}$$
11 $$1 + 5.68T + 11T^{2}$$
13 $$1 - 0.884T + 13T^{2}$$
19 $$1 + 2.71T + 19T^{2}$$
23 $$1 + 5.54T + 23T^{2}$$
29 $$1 + 2.01T + 29T^{2}$$
31 $$1 + 8.73T + 31T^{2}$$
37 $$1 + 2.54T + 37T^{2}$$
41 $$1 + 0.0735T + 41T^{2}$$
43 $$1 + 9.11T + 43T^{2}$$
47 $$1 - 6.75T + 47T^{2}$$
53 $$1 + 6.52T + 53T^{2}$$
59 $$1 - 4.86T + 59T^{2}$$
61 $$1 - 5.04T + 61T^{2}$$
67 $$1 + 4.00T + 67T^{2}$$
71 $$1 - 13.2T + 71T^{2}$$
73 $$1 - 11.9T + 73T^{2}$$
79 $$1 + 8.53T + 79T^{2}$$
83 $$1 - 3.13T + 83T^{2}$$
89 $$1 + 9.82T + 89T^{2}$$
97 $$1 + 8.29T + 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

−7.29512636391638086639523611143, −6.58886703995400330403380229514, −5.80864178529629829712180757618, −5.27348030758025954055858446296, −4.75555273870391425163363613409, −3.94681017900571089554125258238, −3.50596238869972607404731800989, −2.43269046501772608369785954473, −1.77146994999901311887202721471, 0, 1.77146994999901311887202721471, 2.43269046501772608369785954473, 3.50596238869972607404731800989, 3.94681017900571089554125258238, 4.75555273870391425163363613409, 5.27348030758025954055858446296, 5.80864178529629829712180757618, 6.58886703995400330403380229514, 7.29512636391638086639523611143