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  + 26.3·2-s − 164.·3-s + 180.·4-s + 879.·5-s − 4.33e3·6-s + 8.43e3·7-s − 8.73e3·8-s + 7.51e3·9-s + 2.31e4·10-s − 4.44e4·11-s − 2.97e4·12-s − 4.71e4·13-s + 2.21e5·14-s − 1.44e5·15-s − 3.21e5·16-s + 1.06e5·17-s + 1.97e5·18-s − 8.40e5·19-s + 1.58e5·20-s − 1.39e6·21-s − 1.16e6·22-s − 2.20e6·23-s + 1.44e6·24-s − 1.18e6·25-s − 1.24e6·26-s + 2.00e6·27-s + 1.51e6·28-s + ⋯
L(s)  = 1  + 1.16·2-s − 1.17·3-s + 0.351·4-s + 0.628·5-s − 1.36·6-s + 1.32·7-s − 0.753·8-s + 0.381·9-s + 0.731·10-s − 0.914·11-s − 0.413·12-s − 0.458·13-s + 1.54·14-s − 0.739·15-s − 1.22·16-s + 0.308·17-s + 0.444·18-s − 1.47·19-s + 0.221·20-s − 1.56·21-s − 1.06·22-s − 1.64·23-s + 0.886·24-s − 0.604·25-s − 0.532·26-s + 0.726·27-s + 0.466·28-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 \( 1 - 26.3T + 512T^{2} \)
3 \( 1 + 164.T + 1.96e4T^{2} \)
5 \( 1 - 879.T + 1.95e6T^{2} \)
7 \( 1 - 8.43e3T + 4.03e7T^{2} \)
11 \( 1 + 4.44e4T + 2.35e9T^{2} \)
13 \( 1 + 4.71e4T + 1.06e10T^{2} \)
17 \( 1 - 1.06e5T + 1.18e11T^{2} \)
19 \( 1 + 8.40e5T + 3.22e11T^{2} \)
23 \( 1 + 2.20e6T + 1.80e12T^{2} \)
29 \( 1 + 5.29e6T + 1.45e13T^{2} \)
31 \( 1 - 8.55e6T + 2.64e13T^{2} \)
37 \( 1 - 1.40e7T + 1.29e14T^{2} \)
41 \( 1 + 3.26e7T + 3.27e14T^{2} \)
47 \( 1 - 1.82e7T + 1.11e15T^{2} \)
53 \( 1 + 1.58e7T + 3.29e15T^{2} \)
59 \( 1 - 5.21e7T + 8.66e15T^{2} \)
61 \( 1 + 5.19e7T + 1.16e16T^{2} \)
67 \( 1 + 5.12e7T + 2.72e16T^{2} \)
71 \( 1 - 1.33e8T + 4.58e16T^{2} \)
73 \( 1 - 2.51e8T + 5.88e16T^{2} \)
79 \( 1 + 5.19e8T + 1.19e17T^{2} \)
83 \( 1 + 9.09e6T + 1.86e17T^{2} \)
89 \( 1 - 2.78e8T + 3.50e17T^{2} \)
97 \( 1 - 1.06e9T + 7.60e17T^{2} \)
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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

−13.38199340244296194711483597184, −12.21166256534940925885549596832, −11.36486077843884203648433746029, −10.14279563840900747531373393889, −8.167940722276506425508706978778, −6.18684706533089490786234774380, −5.35451689005054477530767853657, −4.42384181082453240730462137719, −2.16456689586200677242994575721, 0, 2.16456689586200677242994575721, 4.42384181082453240730462137719, 5.35451689005054477530767853657, 6.18684706533089490786234774380, 8.167940722276506425508706978778, 10.14279563840900747531373393889, 11.36486077843884203648433746029, 12.21166256534940925885549596832, 13.38199340244296194711483597184

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