# Properties

 Degree 2 Conductor $2 \cdot 19$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s − 2·9-s − 6·11-s + 12-s + 5·13-s + 14-s + 16-s + 3·17-s + 2·18-s + 19-s − 21-s + 6·22-s + 3·23-s − 24-s − 5·25-s − 5·26-s − 5·27-s − 28-s + 9·29-s − 4·31-s − 32-s − 6·33-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 1.80·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.229·19-s − 0.218·21-s + 1.27·22-s + 0.625·23-s − 0.204·24-s − 25-s − 0.980·26-s − 0.962·27-s − 0.188·28-s + 1.67·29-s − 0.718·31-s − 0.176·32-s − 1.04·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$38$$    =    $$2 \cdot 19$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{38} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 38,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.630210$ $L(\frac12)$ $\approx$ $0.630210$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;19\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + T$$
19 $$1 - T$$
good3 $$1 - T + p T^{2}$$
5 $$1 + p T^{2}$$
7 $$1 + T + p T^{2}$$
11 $$1 + 6 T + p T^{2}$$
13 $$1 - 5 T + p T^{2}$$
17 $$1 - 3 T + p T^{2}$$
23 $$1 - 3 T + p T^{2}$$
29 $$1 - 9 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 - 8 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + 3 T + p T^{2}$$
59 $$1 - 9 T + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 - 5 T + p T^{2}$$
71 $$1 + 6 T + p T^{2}$$
73 $$1 + 7 T + p T^{2}$$
79 $$1 + 10 T + p T^{2}$$
83 $$1 + 6 T + p T^{2}$$
89 $$1 + 12 T + p T^{2}$$
97 $$1 + 10 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

−16.22302156666961409355458160527, −15.51354522269081565751677762935, −14.02216922667742784390855478973, −12.90671167977020138547910461547, −11.24807629195878921565630077767, −10.09897479975324709020702728875, −8.684413720393242419277779815238, −7.73591843742988285704805588989, −5.81027585652647529822118508915, −2.99432139105097413830284496966, 2.99432139105097413830284496966, 5.81027585652647529822118508915, 7.73591843742988285704805588989, 8.684413720393242419277779815238, 10.09897479975324709020702728875, 11.24807629195878921565630077767, 12.90671167977020138547910461547, 14.02216922667742784390855478973, 15.51354522269081565751677762935, 16.22302156666961409355458160527