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  − 9.62·2-s + 189.·3-s − 419.·4-s + 636.·5-s − 1.82e3·6-s − 3.28e3·7-s + 8.96e3·8-s + 1.63e4·9-s − 6.12e3·10-s − 7.18e4·11-s − 7.96e4·12-s − 5.64e4·13-s + 3.16e4·14-s + 1.20e5·15-s + 1.28e5·16-s + 2.27e5·17-s − 1.57e5·18-s − 2.58e5·19-s − 2.66e5·20-s − 6.24e5·21-s + 6.92e5·22-s − 2.05e5·23-s + 1.70e6·24-s − 1.54e6·25-s + 5.44e5·26-s − 6.26e5·27-s + 1.37e6·28-s + ⋯
L(s)  = 1  − 0.425·2-s + 1.35·3-s − 0.818·4-s + 0.455·5-s − 0.576·6-s − 0.517·7-s + 0.774·8-s + 0.832·9-s − 0.193·10-s − 1.48·11-s − 1.10·12-s − 0.548·13-s + 0.220·14-s + 0.616·15-s + 0.489·16-s + 0.660·17-s − 0.354·18-s − 0.454·19-s − 0.372·20-s − 0.700·21-s + 0.630·22-s − 0.153·23-s + 1.04·24-s − 0.792·25-s + 0.233·26-s − 0.226·27-s + 0.423·28-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 \( 1 + 9.62T + 512T^{2} \)
3 \( 1 - 189.T + 1.96e4T^{2} \)
5 \( 1 - 636.T + 1.95e6T^{2} \)
7 \( 1 + 3.28e3T + 4.03e7T^{2} \)
11 \( 1 + 7.18e4T + 2.35e9T^{2} \)
13 \( 1 + 5.64e4T + 1.06e10T^{2} \)
17 \( 1 - 2.27e5T + 1.18e11T^{2} \)
19 \( 1 + 2.58e5T + 3.22e11T^{2} \)
23 \( 1 + 2.05e5T + 1.80e12T^{2} \)
29 \( 1 - 5.68e5T + 1.45e13T^{2} \)
31 \( 1 + 6.46e6T + 2.64e13T^{2} \)
37 \( 1 + 1.41e7T + 1.29e14T^{2} \)
41 \( 1 + 1.35e7T + 3.27e14T^{2} \)
47 \( 1 - 2.14e7T + 1.11e15T^{2} \)
53 \( 1 - 3.02e7T + 3.29e15T^{2} \)
59 \( 1 - 1.16e8T + 8.66e15T^{2} \)
61 \( 1 + 1.09e8T + 1.16e16T^{2} \)
67 \( 1 - 1.24e8T + 2.72e16T^{2} \)
71 \( 1 - 2.04e8T + 4.58e16T^{2} \)
73 \( 1 - 1.16e8T + 5.88e16T^{2} \)
79 \( 1 - 4.47e8T + 1.19e17T^{2} \)
83 \( 1 + 5.31e8T + 1.86e17T^{2} \)
89 \( 1 - 2.15e7T + 3.50e17T^{2} \)
97 \( 1 - 9.39e8T + 7.60e17T^{2} \)
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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

−13.51434058387821056822027290831, −12.71152653259957391512929243186, −10.32739010972409485563572669626, −9.572589164153128913158934310820, −8.478716682629762485678444747787, −7.53983275586280920592377246723, −5.30566521076571827611021570563, −3.52883635910927260172430903801, −2.12296593180616741384356879808, 0, 2.12296593180616741384356879808, 3.52883635910927260172430903801, 5.30566521076571827611021570563, 7.53983275586280920592377246723, 8.478716682629762485678444747787, 9.572589164153128913158934310820, 10.32739010972409485563572669626, 12.71152653259957391512929243186, 13.51434058387821056822027290831

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