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  − 34.6·2-s + 96.4·3-s + 690.·4-s + 1.85e3·5-s − 3.34e3·6-s − 1.15e4·7-s − 6.17e3·8-s − 1.03e4·9-s − 6.43e4·10-s + 7.53e4·11-s + 6.66e4·12-s − 1.25e5·13-s + 4.00e5·14-s + 1.79e5·15-s − 1.39e5·16-s + 1.22e5·17-s + 3.59e5·18-s + 4.24e5·19-s + 1.28e6·20-s − 1.11e6·21-s − 2.61e6·22-s − 8.84e5·23-s − 5.96e5·24-s + 1.48e6·25-s + 4.35e6·26-s − 2.90e6·27-s − 7.96e6·28-s + ⋯
L(s)  = 1  − 1.53·2-s + 0.687·3-s + 1.34·4-s + 1.32·5-s − 1.05·6-s − 1.81·7-s − 0.533·8-s − 0.526·9-s − 2.03·10-s + 1.55·11-s + 0.927·12-s − 1.21·13-s + 2.78·14-s + 0.913·15-s − 0.530·16-s + 0.356·17-s + 0.807·18-s + 0.748·19-s + 1.78·20-s − 1.24·21-s − 2.37·22-s − 0.659·23-s − 0.366·24-s + 0.762·25-s + 1.86·26-s − 1.05·27-s − 2.44·28-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 \( 1 + 34.6T + 512T^{2} \)
3 \( 1 - 96.4T + 1.96e4T^{2} \)
5 \( 1 - 1.85e3T + 1.95e6T^{2} \)
7 \( 1 + 1.15e4T + 4.03e7T^{2} \)
11 \( 1 - 7.53e4T + 2.35e9T^{2} \)
13 \( 1 + 1.25e5T + 1.06e10T^{2} \)
17 \( 1 - 1.22e5T + 1.18e11T^{2} \)
19 \( 1 - 4.24e5T + 3.22e11T^{2} \)
23 \( 1 + 8.84e5T + 1.80e12T^{2} \)
29 \( 1 - 2.04e6T + 1.45e13T^{2} \)
31 \( 1 + 6.30e6T + 2.64e13T^{2} \)
37 \( 1 + 1.74e7T + 1.29e14T^{2} \)
41 \( 1 - 2.62e6T + 3.27e14T^{2} \)
47 \( 1 + 2.82e7T + 1.11e15T^{2} \)
53 \( 1 + 6.08e7T + 3.29e15T^{2} \)
59 \( 1 - 1.13e7T + 8.66e15T^{2} \)
61 \( 1 + 7.50e7T + 1.16e16T^{2} \)
67 \( 1 + 1.92e8T + 2.72e16T^{2} \)
71 \( 1 + 3.76e8T + 4.58e16T^{2} \)
73 \( 1 + 1.95e8T + 5.88e16T^{2} \)
79 \( 1 + 3.67e8T + 1.19e17T^{2} \)
83 \( 1 - 4.46e8T + 1.86e17T^{2} \)
89 \( 1 - 7.20e8T + 3.50e17T^{2} \)
97 \( 1 + 1.39e9T + 7.60e17T^{2} \)
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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

−13.56239503292354497888542765230, −12.06539741431703915348540181495, −10.13084786894521049075803175518, −9.500878601733287163041532024278, −8.980396410515265066229368593070, −7.17614991757299431533568559626, −6.07423147975151956391575858013, −3.10807836138298163644508610785, −1.75420719540305403272718199028, 0, 1.75420719540305403272718199028, 3.10807836138298163644508610785, 6.07423147975151956391575858013, 7.17614991757299431533568559626, 8.980396410515265066229368593070, 9.500878601733287163041532024278, 10.13084786894521049075803175518, 12.06539741431703915348540181495, 13.56239503292354497888542765230

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