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  − 41.2·2-s − 186.·3-s + 1.18e3·4-s − 1.99e3·5-s + 7.69e3·6-s − 1.06e4·7-s − 2.77e4·8-s + 1.52e4·9-s + 8.23e4·10-s + 3.62e4·11-s − 2.21e5·12-s + 1.64e5·13-s + 4.40e5·14-s + 3.73e5·15-s + 5.37e5·16-s − 3.97e5·17-s − 6.27e5·18-s + 5.01e5·19-s − 2.37e6·20-s + 1.99e6·21-s − 1.49e6·22-s + 3.42e5·23-s + 5.18e6·24-s + 2.04e6·25-s − 6.78e6·26-s + 8.34e5·27-s − 1.26e7·28-s + ⋯
L(s)  = 1  − 1.82·2-s − 1.33·3-s + 2.31·4-s − 1.43·5-s + 2.42·6-s − 1.68·7-s − 2.39·8-s + 0.773·9-s + 2.60·10-s + 0.747·11-s − 3.08·12-s + 1.59·13-s + 3.06·14-s + 1.90·15-s + 2.04·16-s − 1.15·17-s − 1.40·18-s + 0.882·19-s − 3.31·20-s + 2.24·21-s − 1.36·22-s + 0.255·23-s + 3.19·24-s + 1.04·25-s − 2.91·26-s + 0.302·27-s − 3.89·28-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 \( 1 + 41.2T + 512T^{2} \)
3 \( 1 + 186.T + 1.96e4T^{2} \)
5 \( 1 + 1.99e3T + 1.95e6T^{2} \)
7 \( 1 + 1.06e4T + 4.03e7T^{2} \)
11 \( 1 - 3.62e4T + 2.35e9T^{2} \)
13 \( 1 - 1.64e5T + 1.06e10T^{2} \)
17 \( 1 + 3.97e5T + 1.18e11T^{2} \)
19 \( 1 - 5.01e5T + 3.22e11T^{2} \)
23 \( 1 - 3.42e5T + 1.80e12T^{2} \)
29 \( 1 + 2.05e5T + 1.45e13T^{2} \)
31 \( 1 + 4.44e6T + 2.64e13T^{2} \)
37 \( 1 - 9.29e6T + 1.29e14T^{2} \)
41 \( 1 + 2.39e6T + 3.27e14T^{2} \)
47 \( 1 + 4.72e7T + 1.11e15T^{2} \)
53 \( 1 + 9.18e7T + 3.29e15T^{2} \)
59 \( 1 - 1.40e8T + 8.66e15T^{2} \)
61 \( 1 - 8.99e7T + 1.16e16T^{2} \)
67 \( 1 - 2.51e8T + 2.72e16T^{2} \)
71 \( 1 - 3.30e8T + 4.58e16T^{2} \)
73 \( 1 + 1.51e8T + 5.88e16T^{2} \)
79 \( 1 - 2.48e8T + 1.19e17T^{2} \)
83 \( 1 + 2.98e8T + 1.86e17T^{2} \)
89 \( 1 + 8.14e8T + 3.50e17T^{2} \)
97 \( 1 - 9.42e8T + 7.60e17T^{2} \)
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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

−12.75158346191503642893710099583, −11.42431708838062635483263011756, −11.15192072690357629724764798864, −9.683812276235017425835279565611, −8.526636077594624449019245954450, −6.95999355436907399467148460050, −6.28706142179067292813262899334, −3.54302431425833283025794108926, −0.862673607781792889840048745643, 0, 0.862673607781792889840048745643, 3.54302431425833283025794108926, 6.28706142179067292813262899334, 6.95999355436907399467148460050, 8.526636077594624449019245954450, 9.683812276235017425835279565611, 11.15192072690357629724764798864, 11.42431708838062635483263011756, 12.75158346191503642893710099583

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