# Properties

 Degree $2$ Conductor $1849$ Sign $-1$ Motivic weight $3$ Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 2.53·2-s + 8.01·3-s − 1.58·4-s + 20.7·5-s − 20.2·6-s + 3.06·7-s + 24.2·8-s + 37.2·9-s − 52.4·10-s − 50.3·11-s − 12.7·12-s − 26.3·13-s − 7.75·14-s + 165.·15-s − 48.7·16-s − 23.1·17-s − 94.2·18-s − 135.·19-s − 32.8·20-s + 24.5·21-s + 127.·22-s − 91.8·23-s + 194.·24-s + 304.·25-s + 66.7·26-s + 81.8·27-s − 4.85·28-s + ⋯
 L(s)  = 1 − 0.895·2-s + 1.54·3-s − 0.198·4-s + 1.85·5-s − 1.38·6-s + 0.165·7-s + 1.07·8-s + 1.37·9-s − 1.65·10-s − 1.37·11-s − 0.305·12-s − 0.562·13-s − 0.147·14-s + 2.85·15-s − 0.762·16-s − 0.329·17-s − 1.23·18-s − 1.64·19-s − 0.367·20-s + 0.254·21-s + 1.23·22-s − 0.832·23-s + 1.65·24-s + 2.43·25-s + 0.503·26-s + 0.583·27-s − 0.0327·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1849$$    =    $$43^{2}$$ Sign: $-1$ Motivic weight: $$3$$ Character: $\chi_{1849} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1849,\ (\ :3/2),\ -1)$$

## 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$$F_p(T)$
bad43 $$1$$
good2 $$1 + 2.53T + 8T^{2}$$
3 $$1 - 8.01T + 27T^{2}$$
5 $$1 - 20.7T + 125T^{2}$$
7 $$1 - 3.06T + 343T^{2}$$
11 $$1 + 50.3T + 1.33e3T^{2}$$
13 $$1 + 26.3T + 2.19e3T^{2}$$
17 $$1 + 23.1T + 4.91e3T^{2}$$
19 $$1 + 135.T + 6.85e3T^{2}$$
23 $$1 + 91.8T + 1.21e4T^{2}$$
29 $$1 + 15.5T + 2.43e4T^{2}$$
31 $$1 + 34.9T + 2.97e4T^{2}$$
37 $$1 + 221.T + 5.06e4T^{2}$$
41 $$1 - 206.T + 6.89e4T^{2}$$
47 $$1 + 600.T + 1.03e5T^{2}$$
53 $$1 - 408.T + 1.48e5T^{2}$$
59 $$1 + 740.T + 2.05e5T^{2}$$
61 $$1 - 51.7T + 2.26e5T^{2}$$
67 $$1 - 334.T + 3.00e5T^{2}$$
71 $$1 + 487.T + 3.57e5T^{2}$$
73 $$1 + 722.T + 3.89e5T^{2}$$
79 $$1 + 171.T + 4.93e5T^{2}$$
83 $$1 + 241.T + 5.71e5T^{2}$$
89 $$1 + 333.T + 7.04e5T^{2}$$
97 $$1 - 687.T + 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

−8.652316226089819009995116837714, −8.054330951049633346492337600158, −7.27030402356795367462713199485, −6.22432453568851754391108688798, −5.17565455757664358090681596148, −4.38483972406086336913544154334, −2.95814723554666080438807070795, −2.09442896443710035624456607069, −1.75306264457445927967283020093, 0, 1.75306264457445927967283020093, 2.09442896443710035624456607069, 2.95814723554666080438807070795, 4.38483972406086336913544154334, 5.17565455757664358090681596148, 6.22432453568851754391108688798, 7.27030402356795367462713199485, 8.054330951049633346492337600158, 8.652316226089819009995116837714