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  + 33.4·2-s + 8.30·3-s + 606.·4-s − 752.·5-s + 277.·6-s − 7.16e3·7-s + 3.17e3·8-s − 1.96e4·9-s − 2.51e4·10-s − 3.09e4·11-s + 5.03e3·12-s + 4.18e4·13-s − 2.39e5·14-s − 6.25e3·15-s − 2.04e5·16-s + 1.87e5·17-s − 6.56e5·18-s + 6.68e5·19-s − 4.56e5·20-s − 5.94e4·21-s − 1.03e6·22-s − 2.17e6·23-s + 2.63e4·24-s − 1.38e6·25-s + 1.40e6·26-s − 3.26e5·27-s − 4.34e6·28-s + ⋯
L(s)  = 1  + 1.47·2-s + 0.0591·3-s + 1.18·4-s − 0.538·5-s + 0.0875·6-s − 1.12·7-s + 0.273·8-s − 0.996·9-s − 0.796·10-s − 0.637·11-s + 0.0701·12-s + 0.406·13-s − 1.66·14-s − 0.0318·15-s − 0.780·16-s + 0.543·17-s − 1.47·18-s + 1.17·19-s − 0.638·20-s − 0.0667·21-s − 0.942·22-s − 1.62·23-s + 0.0162·24-s − 0.709·25-s + 0.601·26-s − 0.118·27-s − 1.33·28-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 \( 1 - 33.4T + 512T^{2} \)
3 \( 1 - 8.30T + 1.96e4T^{2} \)
5 \( 1 + 752.T + 1.95e6T^{2} \)
7 \( 1 + 7.16e3T + 4.03e7T^{2} \)
11 \( 1 + 3.09e4T + 2.35e9T^{2} \)
13 \( 1 - 4.18e4T + 1.06e10T^{2} \)
17 \( 1 - 1.87e5T + 1.18e11T^{2} \)
19 \( 1 - 6.68e5T + 3.22e11T^{2} \)
23 \( 1 + 2.17e6T + 1.80e12T^{2} \)
29 \( 1 - 2.40e6T + 1.45e13T^{2} \)
31 \( 1 - 8.05e6T + 2.64e13T^{2} \)
37 \( 1 + 1.64e7T + 1.29e14T^{2} \)
41 \( 1 - 1.13e7T + 3.27e14T^{2} \)
47 \( 1 + 4.31e7T + 1.11e15T^{2} \)
53 \( 1 + 3.13e7T + 3.29e15T^{2} \)
59 \( 1 - 4.51e7T + 8.66e15T^{2} \)
61 \( 1 - 7.25e7T + 1.16e16T^{2} \)
67 \( 1 - 2.83e8T + 2.72e16T^{2} \)
71 \( 1 + 1.90e8T + 4.58e16T^{2} \)
73 \( 1 - 2.75e7T + 5.88e16T^{2} \)
79 \( 1 - 2.64e8T + 1.19e17T^{2} \)
83 \( 1 + 5.42e8T + 1.86e17T^{2} \)
89 \( 1 + 2.40e7T + 3.50e17T^{2} \)
97 \( 1 + 1.30e9T + 7.60e17T^{2} \)
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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

−13.54419517936313883516083666253, −12.31230941379887727548129816477, −11.57279158141761753263340680712, −9.884449548325494563059331847160, −8.132129601329312535267823729722, −6.42545917377148054816736108616, −5.38441148974914961986208700086, −3.73024449667682994690939444576, −2.82851255802725079238259674345, 0, 2.82851255802725079238259674345, 3.73024449667682994690939444576, 5.38441148974914961986208700086, 6.42545917377148054816736108616, 8.132129601329312535267823729722, 9.884449548325494563059331847160, 11.57279158141761753263340680712, 12.31230941379887727548129816477, 13.54419517936313883516083666253

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