# Properties

 Degree 2 Conductor $2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 3-s − 4·7-s + 9-s + 4·11-s − 13-s − 17-s − 8·19-s + 4·21-s − 27-s − 2·29-s − 4·33-s + 10·37-s + 39-s + 10·41-s − 8·43-s + 9·49-s + 51-s − 6·53-s + 8·57-s + 8·59-s + 14·61-s − 4·63-s − 4·67-s + 8·71-s + 6·73-s − 16·77-s + 4·79-s + ⋯
 L(s)  = 1 − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.242·17-s − 1.83·19-s + 0.872·21-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 1.64·37-s + 0.160·39-s + 1.56·41-s − 1.21·43-s + 9/7·49-s + 0.140·51-s − 0.824·53-s + 1.05·57-s + 1.04·59-s + 1.79·61-s − 0.503·63-s − 0.488·67-s + 0.949·71-s + 0.702·73-s − 1.82·77-s + 0.450·79-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$265200$$    =    $$2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{265200} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 265200,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.173186954$ $L(\frac12)$ $\approx$ $1.173186954$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + T$$
5 $$1$$
13 $$1 + T$$
17 $$1 + T$$
good7 $$1 + 4 T + p T^{2}$$
11 $$1 - 4 T + p T^{2}$$
19 $$1 + 8 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + 2 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 - 10 T + p T^{2}$$
41 $$1 - 10 T + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 - 8 T + p T^{2}$$
61 $$1 - 14 T + p T^{2}$$
67 $$1 + 4 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 - 4 T + p T^{2}$$
83 $$1 + 4 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 - 14 T + p T^{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

−12.70278427801761, −12.47771416778450, −11.92150254803620, −11.34166873347079, −11.06512143248047, −10.50782245423058, −9.866104570286672, −9.710018332181685, −9.143597696747124, −8.787117381021653, −8.101938377043321, −7.616169941830722, −6.792697696475040, −6.555901672866440, −6.419453082968214, −5.747619221665675, −5.262597604013642, −4.412363388006993, −4.089003665180022, −3.724917029711294, −2.948278250630313, −2.395999098150488, −1.811291548157965, −0.9188210962621777, −0.3676060917996702, 0.3676060917996702, 0.9188210962621777, 1.811291548157965, 2.395999098150488, 2.948278250630313, 3.724917029711294, 4.089003665180022, 4.412363388006993, 5.262597604013642, 5.747619221665675, 6.419453082968214, 6.555901672866440, 6.792697696475040, 7.616169941830722, 8.101938377043321, 8.787117381021653, 9.143597696747124, 9.710018332181685, 9.866104570286672, 10.50782245423058, 11.06512143248047, 11.34166873347079, 11.92150254803620, 12.47771416778450, 12.70278427801761