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  − 18.4·2-s − 262.·3-s − 170.·4-s − 2.36e3·5-s + 4.84e3·6-s + 7.02e3·7-s + 1.26e4·8-s + 4.90e4·9-s + 4.36e4·10-s − 5.90e4·11-s + 4.46e4·12-s − 8.40e4·13-s − 1.29e5·14-s + 6.19e5·15-s − 1.46e5·16-s + 1.99e5·17-s − 9.06e5·18-s + 2.93e5·19-s + 4.02e5·20-s − 1.84e6·21-s + 1.09e6·22-s + 3.66e5·23-s − 3.30e6·24-s + 3.63e6·25-s + 1.55e6·26-s − 7.68e6·27-s − 1.19e6·28-s + ⋯
L(s)  = 1  − 0.817·2-s − 1.86·3-s − 0.332·4-s − 1.69·5-s + 1.52·6-s + 1.10·7-s + 1.08·8-s + 2.49·9-s + 1.38·10-s − 1.21·11-s + 0.621·12-s − 0.815·13-s − 0.903·14-s + 3.15·15-s − 0.557·16-s + 0.578·17-s − 2.03·18-s + 0.516·19-s + 0.562·20-s − 2.06·21-s + 0.994·22-s + 0.273·23-s − 2.03·24-s + 1.85·25-s + 0.666·26-s − 2.78·27-s − 0.367·28-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 \( 1 + 18.4T + 512T^{2} \)
3 \( 1 + 262.T + 1.96e4T^{2} \)
5 \( 1 + 2.36e3T + 1.95e6T^{2} \)
7 \( 1 - 7.02e3T + 4.03e7T^{2} \)
11 \( 1 + 5.90e4T + 2.35e9T^{2} \)
13 \( 1 + 8.40e4T + 1.06e10T^{2} \)
17 \( 1 - 1.99e5T + 1.18e11T^{2} \)
19 \( 1 - 2.93e5T + 3.22e11T^{2} \)
23 \( 1 - 3.66e5T + 1.80e12T^{2} \)
29 \( 1 + 1.87e6T + 1.45e13T^{2} \)
31 \( 1 - 3.91e6T + 2.64e13T^{2} \)
37 \( 1 - 2.22e7T + 1.29e14T^{2} \)
41 \( 1 - 3.33e6T + 3.27e14T^{2} \)
47 \( 1 + 4.54e7T + 1.11e15T^{2} \)
53 \( 1 - 4.88e7T + 3.29e15T^{2} \)
59 \( 1 - 1.66e8T + 8.66e15T^{2} \)
61 \( 1 + 1.65e8T + 1.16e16T^{2} \)
67 \( 1 - 1.13e7T + 2.72e16T^{2} \)
71 \( 1 + 2.09e8T + 4.58e16T^{2} \)
73 \( 1 - 7.39e6T + 5.88e16T^{2} \)
79 \( 1 - 5.09e8T + 1.19e17T^{2} \)
83 \( 1 + 4.59e8T + 1.86e17T^{2} \)
89 \( 1 + 2.28e8T + 3.50e17T^{2} \)
97 \( 1 + 9.54e8T + 7.60e17T^{2} \)
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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

−12.88508365349008470684414785192, −11.74098319407111090590262775795, −11.10391749287207714211140719381, −10.06632597344989331489419560507, −7.974653511941056439410392415140, −7.41048484416533410605183961359, −5.16207465907814167270559889600, −4.44687149929719658119756565382, −0.932072389055913516288814293645, 0, 0.932072389055913516288814293645, 4.44687149929719658119756565382, 5.16207465907814167270559889600, 7.41048484416533410605183961359, 7.974653511941056439410392415140, 10.06632597344989331489419560507, 11.10391749287207714211140719381, 11.74098319407111090590262775795, 12.88508365349008470684414785192

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