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  + 0.315·2-s + 82.9·3-s − 127.·4-s + 531.·5-s + 26.1·6-s − 349.·7-s − 80.7·8-s + 4.69e3·9-s + 167.·10-s + 3.33e3·11-s − 1.06e4·12-s − 8.58e3·13-s − 110.·14-s + 4.41e4·15-s + 1.63e4·16-s − 2.57e4·17-s + 1.48e3·18-s + 1.69e4·19-s − 6.79e4·20-s − 2.90e4·21-s + 1.05e3·22-s + 8.07e4·23-s − 6.70e3·24-s + 2.04e5·25-s − 2.70e3·26-s + 2.08e5·27-s + 4.47e4·28-s + ⋯
L(s)  = 1  + 0.0279·2-s + 1.77·3-s − 0.999·4-s + 1.90·5-s + 0.0495·6-s − 0.385·7-s − 0.0557·8-s + 2.14·9-s + 0.0530·10-s + 0.754·11-s − 1.77·12-s − 1.08·13-s − 0.0107·14-s + 3.37·15-s + 0.997·16-s − 1.27·17-s + 0.0599·18-s + 0.568·19-s − 1.90·20-s − 0.683·21-s + 0.0210·22-s + 1.38·23-s − 0.0989·24-s + 2.61·25-s − 0.0302·26-s + 2.03·27-s + 0.385·28-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  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(3.43708\)
\(L(\frac12)\)  \(\approx\)  \(3.43708\)
\(L(\frac{9}{2})\)   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 \( 1 - 0.315T + 128T^{2} \)
3 \( 1 - 82.9T + 2.18e3T^{2} \)
5 \( 1 - 531.T + 7.81e4T^{2} \)
7 \( 1 + 349.T + 8.23e5T^{2} \)
11 \( 1 - 3.33e3T + 1.94e7T^{2} \)
13 \( 1 + 8.58e3T + 6.27e7T^{2} \)
17 \( 1 + 2.57e4T + 4.10e8T^{2} \)
19 \( 1 - 1.69e4T + 8.93e8T^{2} \)
23 \( 1 - 8.07e4T + 3.40e9T^{2} \)
29 \( 1 + 6.77e4T + 1.72e10T^{2} \)
31 \( 1 - 1.88e4T + 2.75e10T^{2} \)
37 \( 1 + 1.79e5T + 9.49e10T^{2} \)
41 \( 1 + 6.11e5T + 1.94e11T^{2} \)
47 \( 1 + 1.18e6T + 5.06e11T^{2} \)
53 \( 1 - 5.38e5T + 1.17e12T^{2} \)
59 \( 1 + 1.96e6T + 2.48e12T^{2} \)
61 \( 1 - 2.10e6T + 3.14e12T^{2} \)
67 \( 1 + 8.13e5T + 6.06e12T^{2} \)
71 \( 1 - 1.76e6T + 9.09e12T^{2} \)
73 \( 1 + 3.47e6T + 1.10e13T^{2} \)
79 \( 1 + 3.68e6T + 1.92e13T^{2} \)
83 \( 1 - 3.77e6T + 2.71e13T^{2} \)
89 \( 1 + 2.83e6T + 4.42e13T^{2} \)
97 \( 1 - 4.40e6T + 8.07e13T^{2} \)
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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

−14.26580613645794764599792817278, −13.43652887765336357354855847692, −12.93036365644329662779733018634, −10.00176788422954423633010139137, −9.385316994943992918079487975766, −8.736537338840566664914420887778, −6.84665796552863875515136036201, −4.89401645888586937643917968379, −3.07878366081793291452126047674, −1.71186247415649031714820697881, 1.71186247415649031714820697881, 3.07878366081793291452126047674, 4.89401645888586937643917968379, 6.84665796552863875515136036201, 8.736537338840566664914420887778, 9.385316994943992918079487975766, 10.00176788422954423633010139137, 12.93036365644329662779733018634, 13.43652887765336357354855847692, 14.26580613645794764599792817278

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