Properties

Degree 2
Conductor 43
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s − 4·5-s + 4·6-s + 9-s + 8·10-s + 3·11-s − 4·12-s − 5·13-s + 8·15-s − 4·16-s − 3·17-s − 2·18-s − 2·19-s − 8·20-s − 6·22-s − 23-s + 11·25-s + 10·26-s + 4·27-s − 6·29-s − 16·30-s − 31-s + 8·32-s − 6·33-s + 6·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s + 1/3·9-s + 2.52·10-s + 0.904·11-s − 1.15·12-s − 1.38·13-s + 2.06·15-s − 16-s − 0.727·17-s − 0.471·18-s − 0.458·19-s − 1.78·20-s − 1.27·22-s − 0.208·23-s + 11/5·25-s + 1.96·26-s + 0.769·27-s − 1.11·29-s − 2.92·30-s − 0.179·31-s + 1.41·32-s − 1.04·33-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

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

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 43$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 43$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad43 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−19.58088851446054, −18.71007927047333, −17.47809773099767, −16.82386264281678, −15.98022739565726, −14.78796968128269, −12.38343238787282, −11.51353349230783, −10.79718018756647, −9.206795357679225, −7.864423201644582, −6.828717445793716, −4.494720273981069, 0, 4.494720273981069, 6.828717445793716, 7.864423201644582, 9.206795357679225, 10.79718018756647, 11.51353349230783, 12.38343238787282, 14.78796968128269, 15.98022739565726, 16.82386264281678, 17.47809773099767, 18.71007927047333, 19.58088851446054

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