Properties

Degree 2
Conductor 43
Sign $0.999 - 0.0410i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 31.2i·2-s + 113. i·3-s − 722.·4-s − 959. i·5-s − 3.55e3·6-s + 3.35e3i·7-s − 1.45e4i·8-s − 6.36e3·9-s + 3.00e4·10-s + 5.04e3·11-s − 8.21e4i·12-s − 1.34e4·13-s − 1.04e5·14-s + 1.09e5·15-s + 2.71e5·16-s − 9.36e4·17-s + ⋯
L(s)  = 1  + 1.95i·2-s + 1.40i·3-s − 2.82·4-s − 1.53i·5-s − 2.74·6-s + 1.39i·7-s − 3.55i·8-s − 0.969·9-s + 3.00·10-s + 0.344·11-s − 3.95i·12-s − 0.471·13-s − 2.73·14-s + 2.15·15-s + 4.13·16-s − 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0410i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.999 - 0.0410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.999 - 0.0410i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.999 - 0.0410i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.0792654 + 0.00162917i\)
\(L(\frac12)\)  \(\approx\)  \(0.0792654 + 0.00162917i\)
\(L(5)\)   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 \( 1 + (-3.41e6 + 1.40e5i)T \)
good2 \( 1 - 31.2iT - 256T^{2} \)
3 \( 1 - 113. iT - 6.56e3T^{2} \)
5 \( 1 + 959. iT - 3.90e5T^{2} \)
7 \( 1 - 3.35e3iT - 5.76e6T^{2} \)
11 \( 1 - 5.04e3T + 2.14e8T^{2} \)
13 \( 1 + 1.34e4T + 8.15e8T^{2} \)
17 \( 1 + 9.36e4T + 6.97e9T^{2} \)
19 \( 1 + 7.49e4iT - 1.69e10T^{2} \)
23 \( 1 + 3.09e5T + 7.83e10T^{2} \)
29 \( 1 + 5.68e5iT - 5.00e11T^{2} \)
31 \( 1 - 9.53e5T + 8.52e11T^{2} \)
37 \( 1 - 3.56e3iT - 3.51e12T^{2} \)
41 \( 1 + 2.64e5T + 7.98e12T^{2} \)
47 \( 1 + 8.56e6T + 2.38e13T^{2} \)
53 \( 1 + 3.81e5T + 6.22e13T^{2} \)
59 \( 1 + 9.85e6T + 1.46e14T^{2} \)
61 \( 1 - 1.09e7iT - 1.91e14T^{2} \)
67 \( 1 + 1.57e6T + 4.06e14T^{2} \)
71 \( 1 + 4.72e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.85e7iT - 8.06e14T^{2} \)
79 \( 1 - 2.99e7T + 1.51e15T^{2} \)
83 \( 1 + 5.39e7T + 2.25e15T^{2} \)
89 \( 1 + 1.45e7iT - 3.93e15T^{2} \)
97 \( 1 - 3.45e7T + 7.83e15T^{2} \)
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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

−15.63178101755690426442820680749, −14.89287626130262453286875976413, −13.46145759237487828111259906272, −12.20128632538444716436211834228, −9.615372302014921552518256081308, −9.055836903432271076011325182292, −8.248736824636466069034490251199, −6.07925253410874916244434798741, −4.98820545894514746495095162002, −4.35567461122193415654533454167, 0.03167890074263516998636272734, 1.47419040653978984525823255733, 2.65924719454184162292879364774, 4.02640561826675823896748544893, 6.67768101161337376491709214198, 7.950756034853201879990048587811, 9.937990498851729068332933826263, 10.81291208768388242222012363881, 11.71596467043836512483885173626, 12.87171568847024571610958129099

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