Properties

Degree 2
Conductor 43
Sign $-1$
Motivic weight 9
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 23.7·2-s + 2.00·3-s + 50.3·4-s − 764.·5-s − 47.5·6-s + 4.07e3·7-s + 1.09e4·8-s − 1.96e4·9-s + 1.81e4·10-s + 1.18e4·11-s + 101.·12-s + 1.08e5·13-s − 9.67e4·14-s − 1.53e3·15-s − 2.85e5·16-s + 1.90e5·17-s + 4.66e5·18-s + 7.64e5·19-s − 3.84e4·20-s + 8.18e3·21-s − 2.81e5·22-s − 9.23e5·23-s + 2.19e4·24-s − 1.36e6·25-s − 2.58e6·26-s − 7.89e4·27-s + 2.05e5·28-s + ⋯
L(s)  = 1  − 1.04·2-s + 0.0142·3-s + 0.0983·4-s − 0.546·5-s − 0.0149·6-s + 0.642·7-s + 0.944·8-s − 0.999·9-s + 0.573·10-s + 0.244·11-s + 0.00140·12-s + 1.05·13-s − 0.673·14-s − 0.00781·15-s − 1.08·16-s + 0.553·17-s + 1.04·18-s + 1.34·19-s − 0.0537·20-s + 0.00918·21-s − 0.256·22-s − 0.687·23-s + 0.0135·24-s − 0.701·25-s − 1.10·26-s − 0.0285·27-s + 0.0631·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(9\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ -1)\)
\(L(5)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{11}{2})\)   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 \( 1 + 3.41e6T \)
good2 \( 1 + 23.7T + 512T^{2} \)
3 \( 1 - 2.00T + 1.96e4T^{2} \)
5 \( 1 + 764.T + 1.95e6T^{2} \)
7 \( 1 - 4.07e3T + 4.03e7T^{2} \)
11 \( 1 - 1.18e4T + 2.35e9T^{2} \)
13 \( 1 - 1.08e5T + 1.06e10T^{2} \)
17 \( 1 - 1.90e5T + 1.18e11T^{2} \)
19 \( 1 - 7.64e5T + 3.22e11T^{2} \)
23 \( 1 + 9.23e5T + 1.80e12T^{2} \)
29 \( 1 + 3.16e6T + 1.45e13T^{2} \)
31 \( 1 + 1.00e7T + 2.64e13T^{2} \)
37 \( 1 - 6.60e6T + 1.29e14T^{2} \)
41 \( 1 + 2.24e7T + 3.27e14T^{2} \)
47 \( 1 - 1.96e7T + 1.11e15T^{2} \)
53 \( 1 - 2.18e7T + 3.29e15T^{2} \)
59 \( 1 + 1.05e8T + 8.66e15T^{2} \)
61 \( 1 - 6.39e7T + 1.16e16T^{2} \)
67 \( 1 - 6.09e7T + 2.72e16T^{2} \)
71 \( 1 + 3.75e8T + 4.58e16T^{2} \)
73 \( 1 - 3.38e8T + 5.88e16T^{2} \)
79 \( 1 + 6.12e8T + 1.19e17T^{2} \)
83 \( 1 - 3.98e8T + 1.86e17T^{2} \)
89 \( 1 + 8.15e8T + 3.50e17T^{2} \)
97 \( 1 - 6.93e8T + 7.60e17T^{2} \)
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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

−13.55737041568775818704670351195, −11.74222076142778094144311437511, −10.95295096689937491989407998752, −9.466957563865185193724661409223, −8.392578439652602402513649908041, −7.54795869218909494461902718514, −5.51917719907827706400809653692, −3.71647038354970751297885216811, −1.47861632442151013120781440585, 0, 1.47861632442151013120781440585, 3.71647038354970751297885216811, 5.51917719907827706400809653692, 7.54795869218909494461902718514, 8.392578439652602402513649908041, 9.466957563865185193724661409223, 10.95295096689937491989407998752, 11.74222076142778094144311437511, 13.55737041568775818704670351195

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