Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 7
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.99·2-s + 84.4·3-s − 112.·4-s − 383.·5-s − 337.·6-s + 1.00e3·7-s + 959.·8-s + 4.95e3·9-s + 1.53e3·10-s + 1.62e3·11-s − 9.46e3·12-s + 7.96e3·13-s − 4.01e3·14-s − 3.23e4·15-s + 1.05e4·16-s + 3.64e4·17-s − 1.97e4·18-s − 4.03e4·19-s + 4.29e4·20-s + 8.48e4·21-s − 6.47e3·22-s + 1.33e4·23-s + 8.10e4·24-s + 6.89e4·25-s − 3.18e4·26-s + 2.33e5·27-s − 1.12e5·28-s + ⋯
L(s)  = 1  − 0.353·2-s + 1.80·3-s − 0.875·4-s − 1.37·5-s − 0.638·6-s + 1.10·7-s + 0.662·8-s + 2.26·9-s + 0.484·10-s + 0.367·11-s − 1.58·12-s + 1.00·13-s − 0.390·14-s − 2.47·15-s + 0.641·16-s + 1.79·17-s − 0.799·18-s − 1.34·19-s + 1.20·20-s + 1.99·21-s − 0.129·22-s + 0.228·23-s + 1.19·24-s + 0.882·25-s − 0.354·26-s + 2.28·27-s − 0.968·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(2.22979\)
\(L(\frac12)\)  \(\approx\)  \(2.22979\)
\(L(\frac{9}{2})\)   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 \( 1 + 7.95e4T \)
good2 \( 1 + 3.99T + 128T^{2} \)
3 \( 1 - 84.4T + 2.18e3T^{2} \)
5 \( 1 + 383.T + 7.81e4T^{2} \)
7 \( 1 - 1.00e3T + 8.23e5T^{2} \)
11 \( 1 - 1.62e3T + 1.94e7T^{2} \)
13 \( 1 - 7.96e3T + 6.27e7T^{2} \)
17 \( 1 - 3.64e4T + 4.10e8T^{2} \)
19 \( 1 + 4.03e4T + 8.93e8T^{2} \)
23 \( 1 - 1.33e4T + 3.40e9T^{2} \)
29 \( 1 - 1.19e5T + 1.72e10T^{2} \)
31 \( 1 + 1.03e5T + 2.75e10T^{2} \)
37 \( 1 - 1.91e5T + 9.49e10T^{2} \)
41 \( 1 + 2.40e5T + 1.94e11T^{2} \)
47 \( 1 - 5.02e5T + 5.06e11T^{2} \)
53 \( 1 - 1.09e6T + 1.17e12T^{2} \)
59 \( 1 + 2.43e6T + 2.48e12T^{2} \)
61 \( 1 + 6.20e5T + 3.14e12T^{2} \)
67 \( 1 + 4.73e6T + 6.06e12T^{2} \)
71 \( 1 + 2.81e6T + 9.09e12T^{2} \)
73 \( 1 - 2.78e6T + 1.10e13T^{2} \)
79 \( 1 - 6.00e6T + 1.92e13T^{2} \)
83 \( 1 - 4.68e6T + 2.71e13T^{2} \)
89 \( 1 - 2.94e6T + 4.42e13T^{2} \)
97 \( 1 + 1.41e7T + 8.07e13T^{2} \)
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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

−14.55869229835089675333320815195, −13.60355032696840768432674924544, −12.25960069995517662620450421695, −10.55884523763539074713185143443, −9.002897049007065676973692972884, −8.211641450492993554901775132272, −7.68839465786260536351317386092, −4.40904019845654314294065652746, −3.51708327321286264357112136707, −1.27916372064430856025773472755, 1.27916372064430856025773472755, 3.51708327321286264357112136707, 4.40904019845654314294065652746, 7.68839465786260536351317386092, 8.211641450492993554901775132272, 9.002897049007065676973692972884, 10.55884523763539074713185143443, 12.25960069995517662620450421695, 13.60355032696840768432674924544, 14.55869229835089675333320815195

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