Properties

Degree 2
Conductor 43
Sign $0.652 + 0.757i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25.4i·2-s + 123. i·3-s − 389.·4-s + 710. i·5-s − 3.15e3·6-s − 4.24e3i·7-s − 3.40e3i·8-s − 8.81e3·9-s − 1.80e4·10-s + 1.92e4·11-s − 4.83e4i·12-s − 2.45e4·13-s + 1.07e5·14-s − 8.81e4·15-s − 1.32e4·16-s − 1.42e5·17-s + ⋯
L(s)  = 1  + 1.58i·2-s + 1.53i·3-s − 1.52·4-s + 1.13i·5-s − 2.43·6-s − 1.76i·7-s − 0.831i·8-s − 1.34·9-s − 1.80·10-s + 1.31·11-s − 2.33i·12-s − 0.859·13-s + 2.80·14-s − 1.74·15-s − 0.202·16-s − 1.70·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.652 + 0.757i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.652 + 0.757i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.871680 - 0.399615i\)
\(L(\frac12)\)  \(\approx\)  \(0.871680 - 0.399615i\)
\(L(5)\)   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 \( 1 + (-2.23e6 - 2.59e6i)T \)
good2 \( 1 - 25.4iT - 256T^{2} \)
3 \( 1 - 123. iT - 6.56e3T^{2} \)
5 \( 1 - 710. iT - 3.90e5T^{2} \)
7 \( 1 + 4.24e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.92e4T + 2.14e8T^{2} \)
13 \( 1 + 2.45e4T + 8.15e8T^{2} \)
17 \( 1 + 1.42e5T + 6.97e9T^{2} \)
19 \( 1 - 1.52e5iT - 1.69e10T^{2} \)
23 \( 1 - 1.69e5T + 7.83e10T^{2} \)
29 \( 1 - 3.50e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.67e5T + 8.52e11T^{2} \)
37 \( 1 + 9.65e5iT - 3.51e12T^{2} \)
41 \( 1 + 4.61e6T + 7.98e12T^{2} \)
47 \( 1 - 2.55e6T + 2.38e13T^{2} \)
53 \( 1 + 1.21e5T + 6.22e13T^{2} \)
59 \( 1 - 3.24e6T + 1.46e14T^{2} \)
61 \( 1 - 1.17e7iT - 1.91e14T^{2} \)
67 \( 1 + 3.62e7T + 4.06e14T^{2} \)
71 \( 1 + 1.36e7iT - 6.45e14T^{2} \)
73 \( 1 - 1.61e7iT - 8.06e14T^{2} \)
79 \( 1 - 3.75e7T + 1.51e15T^{2} \)
83 \( 1 - 6.67e7T + 2.25e15T^{2} \)
89 \( 1 - 4.02e7iT - 3.93e15T^{2} \)
97 \( 1 - 3.96e6T + 7.83e15T^{2} \)
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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

−15.05783279023048258823153003029, −14.58873003307903904390750146224, −13.76623886725786842991863895758, −11.10971220339878630207230832984, −10.23814434707237441217146628907, −9.056882349721502554505643543450, −7.31446425940763745825289546210, −6.51520407546007714473659563567, −4.64599661469411787708767311592, −3.75441991550348660260973683199, 0.36134767044291127805143051293, 1.67974681237816226234701169223, 2.51426928698628307377852516584, 4.83342825553265379481988733387, 6.62887914516877475529292476740, 8.799387349138996081127632419030, 9.124050423738358646040167337305, 11.51777829307013139244506996303, 12.05551962235675017627588184640, 12.75361499413393733191964420858

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