# Properties

 Label 2-48e2-1.1-c3-0-53 Degree $2$ Conductor $2304$ Sign $-1$ Analytic cond. $135.940$ Root an. cond. $11.6593$ Motivic weight $3$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 22·5-s − 92·13-s + 104·17-s + 359·25-s − 130·29-s + 396·37-s + 472·41-s − 343·49-s + 518·53-s − 468·61-s + 2.02e3·65-s − 1.09e3·73-s − 2.28e3·85-s + 176·89-s + 594·97-s + 598·101-s + 1.46e3·109-s + 1.32e3·113-s + ⋯
 L(s)  = 1 − 1.96·5-s − 1.96·13-s + 1.48·17-s + 2.87·25-s − 0.832·29-s + 1.75·37-s + 1.79·41-s − 49-s + 1.34·53-s − 0.982·61-s + 3.86·65-s − 1.76·73-s − 2.91·85-s + 0.209·89-s + 0.621·97-s + 0.589·101-s + 1.28·109-s + 1.10·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2304$$    =    $$2^{8} \cdot 3^{2}$$ Sign: $-1$ Analytic conductor: $$135.940$$ Root analytic conductor: $$11.6593$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 2304,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 + 22 T + p^{3} T^{2}$$
7 $$1 + p^{3} T^{2}$$
11 $$1 + p^{3} T^{2}$$
13 $$1 + 92 T + p^{3} T^{2}$$
17 $$1 - 104 T + p^{3} T^{2}$$
19 $$1 + p^{3} T^{2}$$
23 $$1 + p^{3} T^{2}$$
29 $$1 + 130 T + p^{3} T^{2}$$
31 $$1 + p^{3} T^{2}$$
37 $$1 - 396 T + p^{3} T^{2}$$
41 $$1 - 472 T + p^{3} T^{2}$$
43 $$1 + p^{3} T^{2}$$
47 $$1 + p^{3} T^{2}$$
53 $$1 - 518 T + p^{3} T^{2}$$
59 $$1 + p^{3} T^{2}$$
61 $$1 + 468 T + p^{3} T^{2}$$
67 $$1 + p^{3} T^{2}$$
71 $$1 + p^{3} T^{2}$$
73 $$1 + 1098 T + p^{3} T^{2}$$
79 $$1 + p^{3} T^{2}$$
83 $$1 + p^{3} T^{2}$$
89 $$1 - 176 T + p^{3} T^{2}$$
97 $$1 - 594 T + p^{3} 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

−7.894867912411931703206274951198, −7.62876930548703307555672137437, −7.12925052998838309102803294523, −5.85440156707547531551726609467, −4.84109160943709746064046037542, −4.26864956673442495466280777739, −3.35559673055214973587538437005, −2.56865777988317765557524570234, −0.906534888144040192744528763917, 0, 0.906534888144040192744528763917, 2.56865777988317765557524570234, 3.35559673055214973587538437005, 4.26864956673442495466280777739, 4.84109160943709746064046037542, 5.85440156707547531551726609467, 7.12925052998838309102803294523, 7.62876930548703307555672137437, 7.894867912411931703206274951198