Properties

Degree 2
Conductor $ 2 \cdot 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-s − 3-s + 4-s − 4·5-s − 6-s + 3·7-s + 8-s − 2·9-s − 4·10-s + 2·11-s − 12-s − 13-s + 3·14-s + 4·15-s + 16-s + 3·17-s − 2·18-s − 19-s − 4·20-s − 3·21-s + 2·22-s − 23-s − 24-s + 11·25-s − 26-s + 5·27-s + 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s − 1.26·10-s + 0.603·11-s − 0.288·12-s − 0.277·13-s + 0.801·14-s + 1.03·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 0.229·19-s − 0.894·20-s − 0.654·21-s + 0.426·22-s − 0.208·23-s − 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.962·27-s + 0.566·28-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  =  $(2,\ 38,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.819245$
$L(\frac12)$  $\approx$  $0.819245$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
19 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.27742004806018738020850312917, −15.00174295764739089605520124901, −14.37348632970750952303967845812, −12.38825742592724708014161209088, −11.61515083240032291487198991803, −10.98850264170077345648603277341, −8.408486909011940238981299566497, −7.25038064772684706053554512263, −5.28069074518443602737935578596, −3.86057015395193111701613139713, 3.86057015395193111701613139713, 5.28069074518443602737935578596, 7.25038064772684706053554512263, 8.408486909011940238981299566497, 10.98850264170077345648603277341, 11.61515083240032291487198991803, 12.38825742592724708014161209088, 14.37348632970750952303967845812, 15.00174295764739089605520124901, 16.27742004806018738020850312917

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