# Properties

 Degree 2 Conductor $2 \cdot 3 \cdot 5^{2} \cdot 7^{2}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 3.41·11-s + 12-s + 3·13-s + 16-s + 5.82·17-s − 18-s + 1.41·19-s − 3.41·22-s − 0.414·23-s − 24-s − 3·26-s + 27-s + 29-s − 2.41·31-s − 32-s + 3.41·33-s − 5.82·34-s + 36-s + 7.07·37-s − 1.41·38-s + 3·39-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.02·11-s + 0.288·12-s + 0.832·13-s + 0.250·16-s + 1.41·17-s − 0.235·18-s + 0.324·19-s − 0.727·22-s − 0.0863·23-s − 0.204·24-s − 0.588·26-s + 0.192·27-s + 0.185·29-s − 0.433·31-s − 0.176·32-s + 0.594·33-s − 0.999·34-s + 0.166·36-s + 1.16·37-s − 0.229·38-s + 0.480·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$7350$$    =    $$2 \cdot 3 \cdot 5^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{7350} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 7350,\ (\ :1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$2.347745989$$ $$L(\frac12)$$ $$\approx$$ $$2.347745989$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 - T$$
5 $$1$$
7 $$1$$
good11 $$1 - 3.41T + 11T^{2}$$
13 $$1 - 3T + 13T^{2}$$
17 $$1 - 5.82T + 17T^{2}$$
19 $$1 - 1.41T + 19T^{2}$$
23 $$1 + 0.414T + 23T^{2}$$
29 $$1 - T + 29T^{2}$$
31 $$1 + 2.41T + 31T^{2}$$
37 $$1 - 7.07T + 37T^{2}$$
41 $$1 + 5.82T + 41T^{2}$$
43 $$1 - 1.58T + 43T^{2}$$
47 $$1 + 1.65T + 47T^{2}$$
53 $$1 - 12.6T + 53T^{2}$$
59 $$1 - 7.24T + 59T^{2}$$
61 $$1 + 0.171T + 61T^{2}$$
67 $$1 + 15.3T + 67T^{2}$$
71 $$1 + 9.31T + 71T^{2}$$
73 $$1 - 15.0T + 73T^{2}$$
79 $$1 - 1.07T + 79T^{2}$$
83 $$1 + 0.0710T + 83T^{2}$$
89 $$1 - 4T + 89T^{2}$$
97 $$1 + 0.343T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}