Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 7
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.37·2-s − 66.5·3-s − 116.·4-s − 241.·5-s − 224.·6-s − 173.·7-s − 826.·8-s + 2.23e3·9-s − 814.·10-s + 682.·11-s + 7.75e3·12-s + 168.·13-s − 584.·14-s + 1.60e4·15-s + 1.21e4·16-s + 5.56e3·17-s + 7.55e3·18-s − 1.38e4·19-s + 2.81e4·20-s + 1.15e4·21-s + 2.30e3·22-s − 2.46e4·23-s + 5.49e4·24-s − 1.99e4·25-s + 567.·26-s − 3.24e3·27-s + 2.01e4·28-s + ⋯
L(s)  = 1  + 0.298·2-s − 1.42·3-s − 0.910·4-s − 0.862·5-s − 0.424·6-s − 0.190·7-s − 0.570·8-s + 1.02·9-s − 0.257·10-s + 0.154·11-s + 1.29·12-s + 0.0212·13-s − 0.0569·14-s + 1.22·15-s + 0.740·16-s + 0.274·17-s + 0.305·18-s − 0.462·19-s + 0.785·20-s + 0.271·21-s + 0.0461·22-s − 0.422·23-s + 0.811·24-s − 0.255·25-s + 0.00633·26-s − 0.0316·27-s + 0.173·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(0.496703\)
\(L(\frac12)\)  \(\approx\)  \(0.496703\)
\(L(\frac{9}{2})\)   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 \( 1 + 7.95e4T \)
good2 \( 1 - 3.37T + 128T^{2} \)
3 \( 1 + 66.5T + 2.18e3T^{2} \)
5 \( 1 + 241.T + 7.81e4T^{2} \)
7 \( 1 + 173.T + 8.23e5T^{2} \)
11 \( 1 - 682.T + 1.94e7T^{2} \)
13 \( 1 - 168.T + 6.27e7T^{2} \)
17 \( 1 - 5.56e3T + 4.10e8T^{2} \)
19 \( 1 + 1.38e4T + 8.93e8T^{2} \)
23 \( 1 + 2.46e4T + 3.40e9T^{2} \)
29 \( 1 + 3.36e4T + 1.72e10T^{2} \)
31 \( 1 - 2.27e4T + 2.75e10T^{2} \)
37 \( 1 - 1.81e5T + 9.49e10T^{2} \)
41 \( 1 - 3.35e5T + 1.94e11T^{2} \)
47 \( 1 + 7.43e5T + 5.06e11T^{2} \)
53 \( 1 + 1.26e4T + 1.17e12T^{2} \)
59 \( 1 + 8.30e5T + 2.48e12T^{2} \)
61 \( 1 - 2.22e6T + 3.14e12T^{2} \)
67 \( 1 - 3.88e6T + 6.06e12T^{2} \)
71 \( 1 - 3.88e6T + 9.09e12T^{2} \)
73 \( 1 - 5.22e6T + 1.10e13T^{2} \)
79 \( 1 + 2.64e6T + 1.92e13T^{2} \)
83 \( 1 - 6.76e6T + 2.71e13T^{2} \)
89 \( 1 - 3.27e5T + 4.42e13T^{2} \)
97 \( 1 - 5.05e6T + 8.07e13T^{2} \)
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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

−14.44815941293689592546980914022, −12.96574538976420227153819140891, −12.08733264397634016206059855872, −11.12480306491647412994535763258, −9.725919950378911180644992362615, −8.081850405435694298839390372814, −6.34999078225514419469998037061, −5.08325650797735036377069563400, −3.87005227067048405349155023885, −0.53689266807612189750130633243, 0.53689266807612189750130633243, 3.87005227067048405349155023885, 5.08325650797735036377069563400, 6.34999078225514419469998037061, 8.081850405435694298839390372814, 9.725919950378911180644992362615, 11.12480306491647412994535763258, 12.08733264397634016206059855872, 12.96574538976420227153819140891, 14.44815941293689592546980914022

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