Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.50·2-s + 16.8·3-s − 25.7·4-s + 47.4·5-s − 42.2·6-s + 67.4·7-s + 144.·8-s + 42.2·9-s − 118.·10-s + 81.3·11-s − 434.·12-s + 1.05e3·13-s − 168.·14-s + 801.·15-s + 462.·16-s + 251.·17-s − 105.·18-s + 1.61e3·19-s − 1.22e3·20-s + 1.13e3·21-s − 203.·22-s − 32.7·23-s + 2.43e3·24-s − 872.·25-s − 2.64e3·26-s − 3.39e3·27-s − 1.73e3·28-s + ⋯
L(s)  = 1  − 0.441·2-s + 1.08·3-s − 0.804·4-s + 0.849·5-s − 0.478·6-s + 0.520·7-s + 0.797·8-s + 0.173·9-s − 0.375·10-s + 0.202·11-s − 0.871·12-s + 1.73·13-s − 0.229·14-s + 0.919·15-s + 0.452·16-s + 0.210·17-s − 0.0768·18-s + 1.02·19-s − 0.683·20-s + 0.563·21-s − 0.0896·22-s − 0.0129·23-s + 0.864·24-s − 0.279·25-s − 0.767·26-s − 0.895·27-s − 0.418·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.87525\)
\(L(\frac12)\)  \(\approx\)  \(1.87525\)
\(L(\frac{7}{2})\)   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 \( 1 - 1.84e3T \)
good2 \( 1 + 2.50T + 32T^{2} \)
3 \( 1 - 16.8T + 243T^{2} \)
5 \( 1 - 47.4T + 3.12e3T^{2} \)
7 \( 1 - 67.4T + 1.68e4T^{2} \)
11 \( 1 - 81.3T + 1.61e5T^{2} \)
13 \( 1 - 1.05e3T + 3.71e5T^{2} \)
17 \( 1 - 251.T + 1.41e6T^{2} \)
19 \( 1 - 1.61e3T + 2.47e6T^{2} \)
23 \( 1 + 32.7T + 6.43e6T^{2} \)
29 \( 1 + 2.58e3T + 2.05e7T^{2} \)
31 \( 1 + 7.20e3T + 2.86e7T^{2} \)
37 \( 1 + 6.17e3T + 6.93e7T^{2} \)
41 \( 1 - 1.55e4T + 1.15e8T^{2} \)
47 \( 1 + 1.69e3T + 2.29e8T^{2} \)
53 \( 1 + 2.56e4T + 4.18e8T^{2} \)
59 \( 1 - 2.45e4T + 7.14e8T^{2} \)
61 \( 1 - 8.20e3T + 8.44e8T^{2} \)
67 \( 1 + 1.23e4T + 1.35e9T^{2} \)
71 \( 1 - 1.87e4T + 1.80e9T^{2} \)
73 \( 1 + 1.21e4T + 2.07e9T^{2} \)
79 \( 1 + 5.23e4T + 3.07e9T^{2} \)
83 \( 1 - 2.89e4T + 3.93e9T^{2} \)
89 \( 1 - 1.17e5T + 5.58e9T^{2} \)
97 \( 1 + 1.47e5T + 8.58e9T^{2} \)
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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

−14.53021480522552147895318745221, −13.87902087508229585700914862839, −13.11392697388454591463112497374, −11.07200226054915505288133528125, −9.595327485358347353276556440407, −8.848950532587583691227534811289, −7.81771967952932536399312287575, −5.62411278443316773569213887066, −3.66461940834268703106483122238, −1.53976195818631881285740316936, 1.53976195818631881285740316936, 3.66461940834268703106483122238, 5.62411278443316773569213887066, 7.81771967952932536399312287575, 8.848950532587583691227534811289, 9.595327485358347353276556440407, 11.07200226054915505288133528125, 13.11392697388454591463112497374, 13.87902087508229585700914862839, 14.53021480522552147895318745221

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