Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s + 3·5-s − 7-s + 9-s + 3·11-s + 4·12-s − 4·13-s − 6·15-s + 4·16-s − 3·17-s + 19-s − 6·20-s + 2·21-s + 4·25-s + 4·27-s + 2·28-s + 6·29-s − 4·31-s − 6·33-s − 3·35-s − 2·36-s + 2·37-s + 8·39-s − 6·41-s − 43-s − 6·44-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s + 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.15·12-s − 1.10·13-s − 1.54·15-s + 16-s − 0.727·17-s + 0.229·19-s − 1.34·20-s + 0.436·21-s + 4/5·25-s + 0.769·27-s + 0.377·28-s + 1.11·29-s − 0.718·31-s − 1.04·33-s − 0.507·35-s − 1/3·36-s + 0.328·37-s + 1.28·39-s − 0.937·41-s − 0.152·43-s − 0.904·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 19,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.453253$
$L(\frac12)$  $\approx$  $0.453253$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 19$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 19$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad19 \( 1 - T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−18.12139870347555292446209785145, −17.42643155985520821799903376961, −16.70224717326536251258127220715, −14.50114185013199863152077458510, −13.35149180731186594820148200230, −12.06296001644352266090310246809, −10.24320445923116880350231460909, −9.180168549070062598241568188033, −6.39084306102520193435942181609, −5.03912355415234199771433602492, 5.03912355415234199771433602492, 6.39084306102520193435942181609, 9.180168549070062598241568188033, 10.24320445923116880350231460909, 12.06296001644352266090310246809, 13.35149180731186594820148200230, 14.50114185013199863152077458510, 16.70224717326536251258127220715, 17.42643155985520821799903376961, 18.12139870347555292446209785145

Graph of the $Z$-function along the critical line