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  − 42.4·2-s + 107.·3-s + 1.28e3·4-s + 267.·5-s − 4.57e3·6-s + 1.08e3·7-s − 3.28e4·8-s − 8.03e3·9-s − 1.13e4·10-s − 4.97e4·11-s + 1.38e5·12-s + 9.05e4·13-s − 4.59e4·14-s + 2.88e4·15-s + 7.35e5·16-s + 1.39e5·17-s + 3.40e5·18-s − 1.89e5·19-s + 3.43e5·20-s + 1.17e5·21-s + 2.11e6·22-s − 9.49e4·23-s − 3.54e6·24-s − 1.88e6·25-s − 3.84e6·26-s − 2.99e6·27-s + 1.39e6·28-s + ⋯
L(s)  = 1  − 1.87·2-s + 0.769·3-s + 2.51·4-s + 0.191·5-s − 1.44·6-s + 0.170·7-s − 2.83·8-s − 0.408·9-s − 0.358·10-s − 1.02·11-s + 1.93·12-s + 0.879·13-s − 0.319·14-s + 0.147·15-s + 2.80·16-s + 0.405·17-s + 0.764·18-s − 0.334·19-s + 0.480·20-s + 0.131·21-s + 1.92·22-s − 0.0707·23-s − 2.18·24-s − 0.963·25-s − 1.64·26-s − 1.08·27-s + 0.429·28-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 \( 1 + 42.4T + 512T^{2} \)
3 \( 1 - 107.T + 1.96e4T^{2} \)
5 \( 1 - 267.T + 1.95e6T^{2} \)
7 \( 1 - 1.08e3T + 4.03e7T^{2} \)
11 \( 1 + 4.97e4T + 2.35e9T^{2} \)
13 \( 1 - 9.05e4T + 1.06e10T^{2} \)
17 \( 1 - 1.39e5T + 1.18e11T^{2} \)
19 \( 1 + 1.89e5T + 3.22e11T^{2} \)
23 \( 1 + 9.49e4T + 1.80e12T^{2} \)
29 \( 1 + 4.15e6T + 1.45e13T^{2} \)
31 \( 1 - 7.48e6T + 2.64e13T^{2} \)
37 \( 1 - 3.13e6T + 1.29e14T^{2} \)
41 \( 1 + 7.85e6T + 3.27e14T^{2} \)
47 \( 1 + 2.99e7T + 1.11e15T^{2} \)
53 \( 1 - 2.07e7T + 3.29e15T^{2} \)
59 \( 1 + 1.16e8T + 8.66e15T^{2} \)
61 \( 1 + 1.04e8T + 1.16e16T^{2} \)
67 \( 1 + 1.64e8T + 2.72e16T^{2} \)
71 \( 1 + 8.20e7T + 4.58e16T^{2} \)
73 \( 1 + 3.51e8T + 5.88e16T^{2} \)
79 \( 1 - 5.29e8T + 1.19e17T^{2} \)
83 \( 1 + 7.19e8T + 1.86e17T^{2} \)
89 \( 1 - 6.64e8T + 3.50e17T^{2} \)
97 \( 1 - 3.05e8T + 7.60e17T^{2} \)
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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

−13.45697463151290746453098957756, −11.65160486455601167770240839613, −10.56589063179463690117773925601, −9.485991456132261027070002919126, −8.388489731925403627947131978321, −7.73080660589895241282155542731, −6.04276160090495645590271306186, −2.97982993207549499072750346163, −1.70869271523348907545015436592, 0, 1.70869271523348907545015436592, 2.97982993207549499072750346163, 6.04276160090495645590271306186, 7.73080660589895241282155542731, 8.388489731925403627947131978321, 9.485991456132261027070002919126, 10.56589063179463690117773925601, 11.65160486455601167770240839613, 13.45697463151290746453098957756

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