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  − 10.5·2-s + 27.4·3-s + 78.8·4-s + 86.8·5-s − 289.·6-s − 19.8·7-s − 493.·8-s + 512.·9-s − 914.·10-s − 85.3·11-s + 2.16e3·12-s − 229.·13-s + 208.·14-s + 2.38e3·15-s + 2.67e3·16-s + 1.35e3·17-s − 5.40e3·18-s − 2.79e3·19-s + 6.85e3·20-s − 544.·21-s + 898.·22-s − 1.85e3·23-s − 1.35e4·24-s + 4.41e3·25-s + 2.41e3·26-s + 7.42e3·27-s − 1.56e3·28-s + ⋯
L(s)  = 1  − 1.86·2-s + 1.76·3-s + 2.46·4-s + 1.55·5-s − 3.28·6-s − 0.152·7-s − 2.72·8-s + 2.11·9-s − 2.89·10-s − 0.212·11-s + 4.34·12-s − 0.375·13-s + 0.284·14-s + 2.74·15-s + 2.61·16-s + 1.13·17-s − 3.92·18-s − 1.77·19-s + 3.83·20-s − 0.269·21-s + 0.396·22-s − 0.731·23-s − 4.81·24-s + 1.41·25-s + 0.699·26-s + 1.95·27-s − 0.376·28-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  =  \((2,\ 43,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.55218\)
\(L(\frac12)\)  \(\approx\)  \(1.55218\)
\(L(\frac{7}{2})\)   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 \( 1 + 10.5T + 32T^{2} \)
3 \( 1 - 27.4T + 243T^{2} \)
5 \( 1 - 86.8T + 3.12e3T^{2} \)
7 \( 1 + 19.8T + 1.68e4T^{2} \)
11 \( 1 + 85.3T + 1.61e5T^{2} \)
13 \( 1 + 229.T + 3.71e5T^{2} \)
17 \( 1 - 1.35e3T + 1.41e6T^{2} \)
19 \( 1 + 2.79e3T + 2.47e6T^{2} \)
23 \( 1 + 1.85e3T + 6.43e6T^{2} \)
29 \( 1 - 7.31e3T + 2.05e7T^{2} \)
31 \( 1 + 2.93e3T + 2.86e7T^{2} \)
37 \( 1 - 2.57e3T + 6.93e7T^{2} \)
41 \( 1 + 3.53e3T + 1.15e8T^{2} \)
47 \( 1 + 7.06e3T + 2.29e8T^{2} \)
53 \( 1 + 3.85e3T + 4.18e8T^{2} \)
59 \( 1 - 2.79e4T + 7.14e8T^{2} \)
61 \( 1 + 3.92e4T + 8.44e8T^{2} \)
67 \( 1 + 1.48e4T + 1.35e9T^{2} \)
71 \( 1 - 8.95e3T + 1.80e9T^{2} \)
73 \( 1 - 3.51e4T + 2.07e9T^{2} \)
79 \( 1 + 1.32e4T + 3.07e9T^{2} \)
83 \( 1 + 9.81e3T + 3.93e9T^{2} \)
89 \( 1 + 8.71e4T + 5.58e9T^{2} \)
97 \( 1 - 8.39e4T + 8.58e9T^{2} \)
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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

−15.00912791029455002384095018310, −14.06523983565632463473835458185, −12.68562519177398432672809704573, −10.32160990792829855852837183123, −9.813318977465717094574990655173, −8.839698165195743502706891763398, −7.934479134060582057172068095957, −6.48414544058326589576124110429, −2.68021637322697675678592702317, −1.69946400093387293691053616524, 1.69946400093387293691053616524, 2.68021637322697675678592702317, 6.48414544058326589576124110429, 7.934479134060582057172068095957, 8.839698165195743502706891763398, 9.813318977465717094574990655173, 10.32160990792829855852837183123, 12.68562519177398432672809704573, 14.06523983565632463473835458185, 15.00912791029455002384095018310

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