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  − 1.77·2-s − 203.·3-s − 508.·4-s + 1.13e3·5-s + 362.·6-s + 4.32e3·7-s + 1.81e3·8-s + 2.19e4·9-s − 2.02e3·10-s + 3.03e4·11-s + 1.03e5·12-s + 3.41e4·13-s − 7.69e3·14-s − 2.32e5·15-s + 2.57e5·16-s − 5.66e5·17-s − 3.89e4·18-s + 2.46e5·19-s − 5.79e5·20-s − 8.82e5·21-s − 5.40e4·22-s − 1.19e6·23-s − 3.70e5·24-s − 6.55e5·25-s − 6.07e4·26-s − 4.53e5·27-s − 2.20e6·28-s + ⋯
L(s)  = 1  − 0.0786·2-s − 1.45·3-s − 0.993·4-s + 0.815·5-s + 0.114·6-s + 0.680·7-s + 0.156·8-s + 1.11·9-s − 0.0641·10-s + 0.625·11-s + 1.44·12-s + 0.331·13-s − 0.0535·14-s − 1.18·15-s + 0.981·16-s − 1.64·17-s − 0.0875·18-s + 0.434·19-s − 0.810·20-s − 0.989·21-s − 0.0492·22-s − 0.887·23-s − 0.227·24-s − 0.335·25-s − 0.0260·26-s − 0.164·27-s − 0.676·28-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 \( 1 + 1.77T + 512T^{2} \)
3 \( 1 + 203.T + 1.96e4T^{2} \)
5 \( 1 - 1.13e3T + 1.95e6T^{2} \)
7 \( 1 - 4.32e3T + 4.03e7T^{2} \)
11 \( 1 - 3.03e4T + 2.35e9T^{2} \)
13 \( 1 - 3.41e4T + 1.06e10T^{2} \)
17 \( 1 + 5.66e5T + 1.18e11T^{2} \)
19 \( 1 - 2.46e5T + 3.22e11T^{2} \)
23 \( 1 + 1.19e6T + 1.80e12T^{2} \)
29 \( 1 - 7.14e6T + 1.45e13T^{2} \)
31 \( 1 + 5.75e6T + 2.64e13T^{2} \)
37 \( 1 + 8.31e6T + 1.29e14T^{2} \)
41 \( 1 + 4.50e6T + 3.27e14T^{2} \)
47 \( 1 + 2.85e7T + 1.11e15T^{2} \)
53 \( 1 - 2.07e7T + 3.29e15T^{2} \)
59 \( 1 + 3.59e6T + 8.66e15T^{2} \)
61 \( 1 + 5.73e7T + 1.16e16T^{2} \)
67 \( 1 - 1.95e8T + 2.72e16T^{2} \)
71 \( 1 + 4.27e6T + 4.58e16T^{2} \)
73 \( 1 + 1.36e8T + 5.88e16T^{2} \)
79 \( 1 + 3.21e8T + 1.19e17T^{2} \)
83 \( 1 + 5.39e8T + 1.86e17T^{2} \)
89 \( 1 - 2.05e8T + 3.50e17T^{2} \)
97 \( 1 + 1.10e9T + 7.60e17T^{2} \)
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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

−13.38196593093975822103777822610, −12.09288527087925795505529457778, −10.99400820774242247569184781593, −9.855919645960119559476312117478, −8.556600469492015958527622508886, −6.54913641848076058401277656682, −5.41818078912080184372645028598, −4.36935800238721591032898938427, −1.47027173487448859332596641769, 0, 1.47027173487448859332596641769, 4.36935800238721591032898938427, 5.41818078912080184372645028598, 6.54913641848076058401277656682, 8.556600469492015958527622508886, 9.855919645960119559476312117478, 10.99400820774242247569184781593, 12.09288527087925795505529457778, 13.38196593093975822103777822610

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