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  + 7.91·2-s + 12.8·3-s + 30.6·4-s + 79.5·5-s + 101.·6-s − 172.·7-s − 11.0·8-s − 77.1·9-s + 629.·10-s + 452.·11-s + 394.·12-s − 22.7·13-s − 1.36e3·14-s + 1.02e3·15-s − 1.06e3·16-s − 521.·17-s − 610.·18-s + 1.55e3·19-s + 2.43e3·20-s − 2.21e3·21-s + 3.57e3·22-s − 3.46e3·23-s − 142.·24-s + 3.20e3·25-s − 179.·26-s − 4.12e3·27-s − 5.27e3·28-s + ⋯
L(s)  = 1  + 1.39·2-s + 0.826·3-s + 0.956·4-s + 1.42·5-s + 1.15·6-s − 1.32·7-s − 0.0610·8-s − 0.317·9-s + 1.99·10-s + 1.12·11-s + 0.790·12-s − 0.0373·13-s − 1.85·14-s + 1.17·15-s − 1.04·16-s − 0.437·17-s − 0.443·18-s + 0.990·19-s + 1.36·20-s − 1.09·21-s + 1.57·22-s − 1.36·23-s − 0.0504·24-s + 1.02·25-s − 0.0522·26-s − 1.08·27-s − 1.27·28-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  =  \((2,\ 43,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(4.14020\)
\(L(\frac12)\)  \(\approx\)  \(4.14020\)
\(L(\frac{7}{2})\)   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 \( 1 - 7.91T + 32T^{2} \)
3 \( 1 - 12.8T + 243T^{2} \)
5 \( 1 - 79.5T + 3.12e3T^{2} \)
7 \( 1 + 172.T + 1.68e4T^{2} \)
11 \( 1 - 452.T + 1.61e5T^{2} \)
13 \( 1 + 22.7T + 3.71e5T^{2} \)
17 \( 1 + 521.T + 1.41e6T^{2} \)
19 \( 1 - 1.55e3T + 2.47e6T^{2} \)
23 \( 1 + 3.46e3T + 6.43e6T^{2} \)
29 \( 1 - 4.32e3T + 2.05e7T^{2} \)
31 \( 1 + 3.98e3T + 2.86e7T^{2} \)
37 \( 1 - 1.00e4T + 6.93e7T^{2} \)
41 \( 1 + 1.64e4T + 1.15e8T^{2} \)
47 \( 1 - 2.41e4T + 2.29e8T^{2} \)
53 \( 1 - 2.12e4T + 4.18e8T^{2} \)
59 \( 1 + 2.58e4T + 7.14e8T^{2} \)
61 \( 1 - 2.85e4T + 8.44e8T^{2} \)
67 \( 1 - 6.67e4T + 1.35e9T^{2} \)
71 \( 1 + 1.00e4T + 1.80e9T^{2} \)
73 \( 1 - 3.21e4T + 2.07e9T^{2} \)
79 \( 1 + 2.19e4T + 3.07e9T^{2} \)
83 \( 1 + 6.67e4T + 3.93e9T^{2} \)
89 \( 1 - 4.89e4T + 5.58e9T^{2} \)
97 \( 1 - 9.30e4T + 8.58e9T^{2} \)
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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

−14.34658625523888390900322087240, −13.86641445316528797197803290656, −13.09172048736677320018229075088, −11.88586960105509747869882795605, −9.824430344986317449086235401161, −9.036092422463436472160240890261, −6.58325898323565812450751081411, −5.70944105790103814484252456062, −3.69938247848136015406298073891, −2.44971709892072270243620974604, 2.44971709892072270243620974604, 3.69938247848136015406298073891, 5.70944105790103814484252456062, 6.58325898323565812450751081411, 9.036092422463436472160240890261, 9.824430344986317449086235401161, 11.88586960105509747869882795605, 13.09172048736677320018229075088, 13.86641445316528797197803290656, 14.34658625523888390900322087240

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