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  − 15.1·2-s + 48.8·3-s + 101.·4-s + 210.·5-s − 739.·6-s + 1.10e3·7-s + 399.·8-s + 195.·9-s − 3.19e3·10-s − 3.13e3·11-s + 4.96e3·12-s − 5.74e3·13-s − 1.66e4·14-s + 1.02e4·15-s − 1.90e4·16-s + 2.72e4·17-s − 2.95e3·18-s + 5.46e4·19-s + 2.14e4·20-s + 5.37e4·21-s + 4.75e4·22-s + 1.20e4·23-s + 1.94e4·24-s − 3.36e4·25-s + 8.70e4·26-s − 9.72e4·27-s + 1.11e5·28-s + ⋯
L(s)  = 1  − 1.33·2-s + 1.04·3-s + 0.794·4-s + 0.754·5-s − 1.39·6-s + 1.21·7-s + 0.275·8-s + 0.0893·9-s − 1.01·10-s − 0.710·11-s + 0.828·12-s − 0.725·13-s − 1.62·14-s + 0.787·15-s − 1.16·16-s + 1.34·17-s − 0.119·18-s + 1.82·19-s + 0.599·20-s + 1.26·21-s + 0.951·22-s + 0.206·23-s + 0.287·24-s − 0.430·25-s + 0.971·26-s − 0.950·27-s + 0.963·28-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  =  \((2,\ 43,\ (\ :7/2),\ 1)\)
\(L(4)\)  \(\approx\)  \(1.59034\)
\(L(\frac12)\)  \(\approx\)  \(1.59034\)
\(L(\frac{9}{2})\)   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 \( 1 + 15.1T + 128T^{2} \)
3 \( 1 - 48.8T + 2.18e3T^{2} \)
5 \( 1 - 210.T + 7.81e4T^{2} \)
7 \( 1 - 1.10e3T + 8.23e5T^{2} \)
11 \( 1 + 3.13e3T + 1.94e7T^{2} \)
13 \( 1 + 5.74e3T + 6.27e7T^{2} \)
17 \( 1 - 2.72e4T + 4.10e8T^{2} \)
19 \( 1 - 5.46e4T + 8.93e8T^{2} \)
23 \( 1 - 1.20e4T + 3.40e9T^{2} \)
29 \( 1 - 1.32e5T + 1.72e10T^{2} \)
31 \( 1 - 2.81e5T + 2.75e10T^{2} \)
37 \( 1 + 2.14e5T + 9.49e10T^{2} \)
41 \( 1 - 3.02e5T + 1.94e11T^{2} \)
47 \( 1 - 2.35e5T + 5.06e11T^{2} \)
53 \( 1 - 5.50e4T + 1.17e12T^{2} \)
59 \( 1 - 1.50e5T + 2.48e12T^{2} \)
61 \( 1 + 1.65e6T + 3.14e12T^{2} \)
67 \( 1 - 1.76e6T + 6.06e12T^{2} \)
71 \( 1 - 3.57e6T + 9.09e12T^{2} \)
73 \( 1 + 4.30e5T + 1.10e13T^{2} \)
79 \( 1 + 7.44e6T + 1.92e13T^{2} \)
83 \( 1 - 5.50e6T + 2.71e13T^{2} \)
89 \( 1 + 3.53e6T + 4.42e13T^{2} \)
97 \( 1 + 2.08e6T + 8.07e13T^{2} \)
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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

−14.26155476277690717688199424488, −13.76266421742948232031935062750, −11.77233206605313235740506000735, −10.23677653655894626914419363866, −9.470940249041327014086019313859, −8.191587433724598217882546689524, −7.60965295327788226226358776545, −5.17699037487744881173449102474, −2.62864437113062029615022509759, −1.24170678859237207613617355618, 1.24170678859237207613617355618, 2.62864437113062029615022509759, 5.17699037487744881173449102474, 7.60965295327788226226358776545, 8.191587433724598217882546689524, 9.470940249041327014086019313859, 10.23677653655894626914419363866, 11.77233206605313235740506000735, 13.76266421742948232031935062750, 14.26155476277690717688199424488

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