# Properties

 Degree 2 Conductor $2^{3} \cdot 751$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.08·3-s − 3.00·5-s + 4.90·7-s + 6.52·9-s − 0.849·11-s + 3.14·13-s − 9.27·15-s + 0.172·17-s − 5.46·19-s + 15.1·21-s − 1.77·23-s + 4.03·25-s + 10.8·27-s + 4.39·29-s − 1.76·31-s − 2.62·33-s − 14.7·35-s + 7.37·37-s + 9.69·39-s + 7.48·41-s + 10.5·43-s − 19.6·45-s − 12.9·47-s + 17.0·49-s + 0.532·51-s + 4.69·53-s + 2.55·55-s + ⋯
 L(s)  = 1 + 1.78·3-s − 1.34·5-s + 1.85·7-s + 2.17·9-s − 0.256·11-s + 0.871·13-s − 2.39·15-s + 0.0418·17-s − 1.25·19-s + 3.30·21-s − 0.369·23-s + 0.806·25-s + 2.09·27-s + 0.815·29-s − 0.316·31-s − 0.456·33-s − 2.49·35-s + 1.21·37-s + 1.55·39-s + 1.16·41-s + 1.61·43-s − 2.92·45-s − 1.88·47-s + 2.44·49-s + 0.0746·51-s + 0.644·53-s + 0.344·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$6008$$    =    $$2^{3} \cdot 751$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{6008} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 6008,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $4.153672936$ $L(\frac12)$ $\approx$ $4.153672936$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;751\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;751\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
751 $$1 + T$$
good3 $$1 - 3.08T + 3T^{2}$$
5 $$1 + 3.00T + 5T^{2}$$
7 $$1 - 4.90T + 7T^{2}$$
11 $$1 + 0.849T + 11T^{2}$$
13 $$1 - 3.14T + 13T^{2}$$
17 $$1 - 0.172T + 17T^{2}$$
19 $$1 + 5.46T + 19T^{2}$$
23 $$1 + 1.77T + 23T^{2}$$
29 $$1 - 4.39T + 29T^{2}$$
31 $$1 + 1.76T + 31T^{2}$$
37 $$1 - 7.37T + 37T^{2}$$
41 $$1 - 7.48T + 41T^{2}$$
43 $$1 - 10.5T + 43T^{2}$$
47 $$1 + 12.9T + 47T^{2}$$
53 $$1 - 4.69T + 53T^{2}$$
59 $$1 - 6.35T + 59T^{2}$$
61 $$1 - 7.33T + 61T^{2}$$
67 $$1 + 12.2T + 67T^{2}$$
71 $$1 + 4.18T + 71T^{2}$$
73 $$1 - 3.73T + 73T^{2}$$
79 $$1 + 13.9T + 79T^{2}$$
83 $$1 - 0.877T + 83T^{2}$$
89 $$1 + 7.18T + 89T^{2}$$
97 $$1 - 14.1T + 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

−8.037573371948077856408735596146, −7.77433834982207011994777508602, −7.14594147893735175712908783527, −5.99642819384790352671517221309, −4.71734788941075346501871271486, −4.24184821618978066192615319868, −3.79140332668595695325065230841, −2.74690203935553951623502157222, −2.01731761645240002536306957534, −1.05623125550272660015074438502, 1.05623125550272660015074438502, 2.01731761645240002536306957534, 2.74690203935553951623502157222, 3.79140332668595695325065230841, 4.24184821618978066192615319868, 4.71734788941075346501871271486, 5.99642819384790352671517221309, 7.14594147893735175712908783527, 7.77433834982207011994777508602, 8.037573371948077856408735596146