# Properties

 Label 2-48e2-1.1-c3-0-12 Degree $2$ Conductor $2304$ Sign $1$ Analytic cond. $135.940$ Root an. cond. $11.6593$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 3.46·5-s − 24.2·7-s − 48·11-s + 41.5·13-s − 54·17-s − 4·19-s + 173.·23-s − 113·25-s − 162.·29-s + 58.8·31-s + 84·35-s − 325.·37-s − 294·41-s + 188·43-s − 505.·47-s + 245·49-s − 744.·53-s + 166.·55-s − 252·59-s − 90.0·61-s − 144·65-s + 628·67-s − 6.92·71-s + 1.00e3·73-s + 1.16e3·77-s − 1.34e3·79-s + 720·83-s + ⋯
 L(s)  = 1 − 0.309·5-s − 1.30·7-s − 1.31·11-s + 0.886·13-s − 0.770·17-s − 0.0482·19-s + 1.57·23-s − 0.904·25-s − 1.04·29-s + 0.341·31-s + 0.405·35-s − 1.44·37-s − 1.11·41-s + 0.666·43-s − 1.56·47-s + 0.714·49-s − 1.93·53-s + 0.407·55-s − 0.556·59-s − 0.189·61-s − 0.274·65-s + 1.14·67-s − 0.0115·71-s + 1.61·73-s + 1.72·77-s − 1.90·79-s + 0.952·83-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: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2304,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.6465933606$$ $$L(\frac12)$$ $$\approx$$ $$0.6465933606$$ $$L(\frac{5}{2})$$ 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 + 3.46T + 125T^{2}$$
7 $$1 + 24.2T + 343T^{2}$$
11 $$1 + 48T + 1.33e3T^{2}$$
13 $$1 - 41.5T + 2.19e3T^{2}$$
17 $$1 + 54T + 4.91e3T^{2}$$
19 $$1 + 4T + 6.85e3T^{2}$$
23 $$1 - 173.T + 1.21e4T^{2}$$
29 $$1 + 162.T + 2.43e4T^{2}$$
31 $$1 - 58.8T + 2.97e4T^{2}$$
37 $$1 + 325.T + 5.06e4T^{2}$$
41 $$1 + 294T + 6.89e4T^{2}$$
43 $$1 - 188T + 7.95e4T^{2}$$
47 $$1 + 505.T + 1.03e5T^{2}$$
53 $$1 + 744.T + 1.48e5T^{2}$$
59 $$1 + 252T + 2.05e5T^{2}$$
61 $$1 + 90.0T + 2.26e5T^{2}$$
67 $$1 - 628T + 3.00e5T^{2}$$
71 $$1 + 6.92T + 3.57e5T^{2}$$
73 $$1 - 1.00e3T + 3.89e5T^{2}$$
79 $$1 + 1.34e3T + 4.93e5T^{2}$$
83 $$1 - 720T + 5.71e5T^{2}$$
89 $$1 + 1.48e3T + 7.04e5T^{2}$$
97 $$1 - 1.82e3T + 9.12e5T^{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

−8.655430955944584362729689025672, −7.919340838308036289195149600747, −7.01805705152733811111974488598, −6.41693681530330771086557399053, −5.53526900193583728001193936892, −4.70375282267492794202573003259, −3.51566297451460690966381109263, −3.06549719186210529068662925749, −1.84857299912999640188911938285, −0.34231743596383099664392146484, 0.34231743596383099664392146484, 1.84857299912999640188911938285, 3.06549719186210529068662925749, 3.51566297451460690966381109263, 4.70375282267492794202573003259, 5.53526900193583728001193936892, 6.41693681530330771086557399053, 7.01805705152733811111974488598, 7.919340838308036289195149600747, 8.655430955944584362729689025672