# Properties

 Label 2-48e2-4.3-c2-0-23 Degree $2$ Conductor $2304$ Sign $1$ Analytic cond. $62.7794$ Root an. cond. $7.92334$ Motivic weight $2$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 6·5-s − 24·13-s + 16·17-s + 11·25-s − 42·29-s + 24·37-s − 80·41-s + 49·49-s − 90·53-s + 120·61-s + 144·65-s + 110·73-s − 96·85-s − 160·89-s + 130·97-s + 198·101-s − 120·109-s + 224·113-s + ⋯
 L(s)  = 1 − 6/5·5-s − 1.84·13-s + 0.941·17-s + 0.439·25-s − 1.44·29-s + 0.648·37-s − 1.95·41-s + 49-s − 1.69·53-s + 1.96·61-s + 2.21·65-s + 1.50·73-s − 1.12·85-s − 1.79·89-s + 1.34·97-s + 1.96·101-s − 1.10·109-s + 1.98·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $1$ Analytic conductor: $$62.7794$$ Root analytic conductor: $$7.92334$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: $\chi_{2304} (1279, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2304,\ (\ :1),\ 1)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.8944955646$$ $$L(\frac12)$$ $$\approx$$ $$0.8944955646$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + 6 T + p^{2} T^{2}$$
7 $$( 1 - p T )( 1 + p T )$$
11 $$( 1 - p T )( 1 + p T )$$
13 $$1 + 24 T + p^{2} T^{2}$$
17 $$1 - 16 T + p^{2} T^{2}$$
19 $$( 1 - p T )( 1 + p T )$$
23 $$( 1 - p T )( 1 + p T )$$
29 $$1 + 42 T + p^{2} T^{2}$$
31 $$( 1 - p T )( 1 + p T )$$
37 $$1 - 24 T + p^{2} T^{2}$$
41 $$1 + 80 T + p^{2} T^{2}$$
43 $$( 1 - p T )( 1 + p T )$$
47 $$( 1 - p T )( 1 + p T )$$
53 $$1 + 90 T + p^{2} T^{2}$$
59 $$( 1 - p T )( 1 + p T )$$
61 $$1 - 120 T + p^{2} T^{2}$$
67 $$( 1 - p T )( 1 + p T )$$
71 $$( 1 - p T )( 1 + p T )$$
73 $$1 - 110 T + p^{2} T^{2}$$
79 $$( 1 - p T )( 1 + p T )$$
83 $$( 1 - p T )( 1 + p T )$$
89 $$1 + 160 T + p^{2} T^{2}$$
97 $$1 - 130 T + p^{2} T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−8.731931215410081020618166246343, −7.81261977694033713446807476831, −7.50728947481209324457190485999, −6.71684368489088035558960355535, −5.49638032536542519672477746773, −4.83922007093774589206459198684, −3.91096068054447725702859093488, −3.14429910710236381825539193641, −2.01417992920065883921986664188, −0.47102575647765960743785862222, 0.47102575647765960743785862222, 2.01417992920065883921986664188, 3.14429910710236381825539193641, 3.91096068054447725702859093488, 4.83922007093774589206459198684, 5.49638032536542519672477746773, 6.71684368489088035558960355535, 7.50728947481209324457190485999, 7.81261977694033713446807476831, 8.731931215410081020618166246343