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  + 23.8·2-s + 64.8·3-s + 57.6·4-s − 578.·5-s + 1.54e3·6-s − 413.·7-s − 1.08e4·8-s − 1.54e4·9-s − 1.38e4·10-s + 4.09e4·11-s + 3.73e3·12-s − 1.54e5·13-s − 9.85e3·14-s − 3.75e4·15-s − 2.88e5·16-s − 3.70e4·17-s − 3.69e5·18-s − 5.72e5·19-s − 3.33e4·20-s − 2.67e4·21-s + 9.78e5·22-s + 1.53e6·23-s − 7.02e5·24-s − 1.61e6·25-s − 3.69e6·26-s − 2.27e6·27-s − 2.38e4·28-s + ⋯
L(s)  = 1  + 1.05·2-s + 0.462·3-s + 0.112·4-s − 0.414·5-s + 0.487·6-s − 0.0650·7-s − 0.935·8-s − 0.786·9-s − 0.436·10-s + 0.844·11-s + 0.0520·12-s − 1.50·13-s − 0.0685·14-s − 0.191·15-s − 1.09·16-s − 0.107·17-s − 0.829·18-s − 1.00·19-s − 0.0466·20-s − 0.0300·21-s + 0.890·22-s + 1.14·23-s − 0.432·24-s − 0.828·25-s − 1.58·26-s − 0.825·27-s − 0.00732·28-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 \( 1 - 23.8T + 512T^{2} \)
3 \( 1 - 64.8T + 1.96e4T^{2} \)
5 \( 1 + 578.T + 1.95e6T^{2} \)
7 \( 1 + 413.T + 4.03e7T^{2} \)
11 \( 1 - 4.09e4T + 2.35e9T^{2} \)
13 \( 1 + 1.54e5T + 1.06e10T^{2} \)
17 \( 1 + 3.70e4T + 1.18e11T^{2} \)
19 \( 1 + 5.72e5T + 3.22e11T^{2} \)
23 \( 1 - 1.53e6T + 1.80e12T^{2} \)
29 \( 1 - 5.10e6T + 1.45e13T^{2} \)
31 \( 1 + 6.63e6T + 2.64e13T^{2} \)
37 \( 1 - 1.17e7T + 1.29e14T^{2} \)
41 \( 1 + 1.86e7T + 3.27e14T^{2} \)
47 \( 1 + 4.17e7T + 1.11e15T^{2} \)
53 \( 1 - 1.09e8T + 3.29e15T^{2} \)
59 \( 1 - 1.08e8T + 8.66e15T^{2} \)
61 \( 1 - 8.96e7T + 1.16e16T^{2} \)
67 \( 1 + 2.02e8T + 2.72e16T^{2} \)
71 \( 1 - 1.86e8T + 4.58e16T^{2} \)
73 \( 1 - 2.88e8T + 5.88e16T^{2} \)
79 \( 1 - 1.54e8T + 1.19e17T^{2} \)
83 \( 1 - 1.44e8T + 1.86e17T^{2} \)
89 \( 1 + 5.50e8T + 3.50e17T^{2} \)
97 \( 1 + 1.15e9T + 7.60e17T^{2} \)
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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

−13.49673941381605663449109709767, −12.36697746888991201477114770343, −11.44690474561391734751437769271, −9.543940186570799767269206135563, −8.404983107990047435461425680516, −6.70938779727046902196556335880, −5.15619612464037689435341636314, −3.87890453550281960566577088392, −2.58181485422625611206236865058, 0, 2.58181485422625611206236865058, 3.87890453550281960566577088392, 5.15619612464037689435341636314, 6.70938779727046902196556335880, 8.404983107990047435461425680516, 9.543940186570799767269206135563, 11.44690474561391734751437769271, 12.36697746888991201477114770343, 13.49673941381605663449109709767

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