Properties

Degree 2
Conductor 43
Sign $-1$
Motivic weight 9
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.0·2-s + 34.0·3-s − 285.·4-s + 1.87e3·5-s + 512.·6-s − 7.04e3·7-s − 1.20e4·8-s − 1.85e4·9-s + 2.82e4·10-s − 2.47e3·11-s − 9.69e3·12-s + 1.03e5·13-s − 1.06e5·14-s + 6.38e4·15-s − 3.49e4·16-s − 5.29e5·17-s − 2.79e5·18-s − 4.31e5·19-s − 5.35e5·20-s − 2.39e5·21-s − 3.73e4·22-s − 1.00e5·23-s − 4.08e5·24-s + 1.56e6·25-s + 1.55e6·26-s − 1.29e6·27-s + 2.00e6·28-s + ⋯
L(s)  = 1  + 0.665·2-s + 0.242·3-s − 0.556·4-s + 1.34·5-s + 0.161·6-s − 1.10·7-s − 1.03·8-s − 0.941·9-s + 0.894·10-s − 0.0510·11-s − 0.134·12-s + 1.00·13-s − 0.737·14-s + 0.325·15-s − 0.133·16-s − 1.53·17-s − 0.626·18-s − 0.760·19-s − 0.747·20-s − 0.268·21-s − 0.0339·22-s − 0.0749·23-s − 0.251·24-s + 0.803·25-s + 0.666·26-s − 0.470·27-s + 0.617·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(9\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ -1)\)
\(L(5)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{11}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 + 3.41e6T \)
good2 \( 1 - 15.0T + 512T^{2} \)
3 \( 1 - 34.0T + 1.96e4T^{2} \)
5 \( 1 - 1.87e3T + 1.95e6T^{2} \)
7 \( 1 + 7.04e3T + 4.03e7T^{2} \)
11 \( 1 + 2.47e3T + 2.35e9T^{2} \)
13 \( 1 - 1.03e5T + 1.06e10T^{2} \)
17 \( 1 + 5.29e5T + 1.18e11T^{2} \)
19 \( 1 + 4.31e5T + 3.22e11T^{2} \)
23 \( 1 + 1.00e5T + 1.80e12T^{2} \)
29 \( 1 + 7.09e6T + 1.45e13T^{2} \)
31 \( 1 + 3.27e6T + 2.64e13T^{2} \)
37 \( 1 + 4.41e6T + 1.29e14T^{2} \)
41 \( 1 - 1.81e7T + 3.27e14T^{2} \)
47 \( 1 - 5.52e7T + 1.11e15T^{2} \)
53 \( 1 + 8.53e6T + 3.29e15T^{2} \)
59 \( 1 + 1.92e7T + 8.66e15T^{2} \)
61 \( 1 + 9.01e6T + 1.16e16T^{2} \)
67 \( 1 - 1.40e8T + 2.72e16T^{2} \)
71 \( 1 - 2.63e8T + 4.58e16T^{2} \)
73 \( 1 + 1.09e8T + 5.88e16T^{2} \)
79 \( 1 + 6.39e7T + 1.19e17T^{2} \)
83 \( 1 - 4.71e8T + 1.86e17T^{2} \)
89 \( 1 + 1.07e9T + 3.50e17T^{2} \)
97 \( 1 + 2.81e8T + 7.60e17T^{2} \)
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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

−13.37738714472998474731786260934, −12.83786822040773987254004134609, −10.96279232640493156797588260158, −9.420103992600823713078162375268, −8.826299670333874658772106265025, −6.35344794109548434155918003221, −5.61806596519482826369821693610, −3.77904499124625568549472568998, −2.35468996523677705203841536537, 0, 2.35468996523677705203841536537, 3.77904499124625568549472568998, 5.61806596519482826369821693610, 6.35344794109548434155918003221, 8.826299670333874658772106265025, 9.420103992600823713078162375268, 10.96279232640493156797588260158, 12.83786822040773987254004134609, 13.37738714472998474731786260934

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