Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 43^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s + 2·5-s − 6-s + 7-s − 3·8-s + 9-s + 2·10-s + 4·11-s + 12-s − 2·13-s + 14-s − 2·15-s − 16-s − 6·17-s + 18-s − 4·19-s − 2·20-s − 21-s + 4·22-s + 3·24-s − 25-s − 2·26-s − 27-s − 28-s + 2·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s + 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 38829 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 38829 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(38829\)    =    \(3 \cdot 7 \cdot 43^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{38829} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 38829,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;43\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;43\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
7 \( 1 - T \)
43 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.87229593605596, −14.38363502129932, −14.16431183405297, −13.33599628516857, −13.20328314324205, −12.54459341092978, −12.00641725750365, −11.53482625304841, −11.05540030789862, −10.17410700948919, −9.974528973163842, −9.130026938838398, −8.843161583996257, −8.341683930454477, −7.275670648713839, −6.727543837140484, −6.246540222883262, −5.779530448653858, −5.109375516680978, −4.599915887999517, −4.114516077025931, −3.520399151306706, −2.387456749057356, −1.995632233185945, −0.9809192731234065, 0, 0.9809192731234065, 1.995632233185945, 2.387456749057356, 3.520399151306706, 4.114516077025931, 4.599915887999517, 5.109375516680978, 5.779530448653858, 6.246540222883262, 6.727543837140484, 7.275670648713839, 8.341683930454477, 8.843161583996257, 9.130026938838398, 9.974528973163842, 10.17410700948919, 11.05540030789862, 11.53482625304841, 12.00641725750365, 12.54459341092978, 13.20328314324205, 13.33599628516857, 14.16431183405297, 14.38363502129932, 14.87229593605596

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