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  − 4.38·2-s + 117.·3-s − 492.·4-s − 1.23e3·5-s − 517.·6-s + 1.25e4·7-s + 4.40e3·8-s − 5.77e3·9-s + 5.42e3·10-s + 2.79e4·11-s − 5.81e4·12-s − 1.39e5·13-s − 5.50e4·14-s − 1.45e5·15-s + 2.32e5·16-s − 4.08e5·17-s + 2.53e4·18-s − 1.57e5·19-s + 6.09e5·20-s + 1.47e6·21-s − 1.22e5·22-s − 7.30e5·23-s + 5.19e5·24-s − 4.22e5·25-s + 6.12e5·26-s − 3.00e6·27-s − 6.18e6·28-s + ⋯
L(s)  = 1  − 0.193·2-s + 0.840·3-s − 0.962·4-s − 0.885·5-s − 0.162·6-s + 1.97·7-s + 0.380·8-s − 0.293·9-s + 0.171·10-s + 0.574·11-s − 0.808·12-s − 1.35·13-s − 0.382·14-s − 0.744·15-s + 0.888·16-s − 1.18·17-s + 0.0568·18-s − 0.277·19-s + 0.851·20-s + 1.66·21-s − 0.111·22-s − 0.544·23-s + 0.319·24-s − 0.216·25-s + 0.262·26-s − 1.08·27-s − 1.90·28-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 \( 1 + 4.38T + 512T^{2} \)
3 \( 1 - 117.T + 1.96e4T^{2} \)
5 \( 1 + 1.23e3T + 1.95e6T^{2} \)
7 \( 1 - 1.25e4T + 4.03e7T^{2} \)
11 \( 1 - 2.79e4T + 2.35e9T^{2} \)
13 \( 1 + 1.39e5T + 1.06e10T^{2} \)
17 \( 1 + 4.08e5T + 1.18e11T^{2} \)
19 \( 1 + 1.57e5T + 3.22e11T^{2} \)
23 \( 1 + 7.30e5T + 1.80e12T^{2} \)
29 \( 1 + 6.14e6T + 1.45e13T^{2} \)
31 \( 1 - 6.18e6T + 2.64e13T^{2} \)
37 \( 1 + 2.16e7T + 1.29e14T^{2} \)
41 \( 1 - 2.42e7T + 3.27e14T^{2} \)
47 \( 1 + 3.21e7T + 1.11e15T^{2} \)
53 \( 1 + 6.96e7T + 3.29e15T^{2} \)
59 \( 1 + 1.12e8T + 8.66e15T^{2} \)
61 \( 1 - 8.03e7T + 1.16e16T^{2} \)
67 \( 1 + 5.78e7T + 2.72e16T^{2} \)
71 \( 1 - 3.11e8T + 4.58e16T^{2} \)
73 \( 1 + 3.23e7T + 5.88e16T^{2} \)
79 \( 1 - 2.36e8T + 1.19e17T^{2} \)
83 \( 1 - 4.61e8T + 1.86e17T^{2} \)
89 \( 1 - 6.16e8T + 3.50e17T^{2} \)
97 \( 1 + 1.00e9T + 7.60e17T^{2} \)
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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

−13.78537432959384790266009763706, −12.10055967743327595125997672199, −11.06498783595345299267102378862, −9.297851097909117396307909190142, −8.295151624828064908093772327819, −7.66649351942361388657516718381, −4.96306054166819861671056525901, −4.00311384923013249688251675871, −1.94900829879763215761899596489, 0, 1.94900829879763215761899596489, 4.00311384923013249688251675871, 4.96306054166819861671056525901, 7.66649351942361388657516718381, 8.295151624828064908093772327819, 9.297851097909117396307909190142, 11.06498783595345299267102378862, 12.10055967743327595125997672199, 13.78537432959384790266009763706

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