Properties

Degree 2
Conductor 43
Sign $-0.584 + 0.811i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.15i·2-s + 11.7i·3-s + 189.·4-s − 1.15e3i·5-s + 96.1·6-s + 1.56e3i·7-s − 3.63e3i·8-s + 6.42e3·9-s − 9.40e3·10-s − 1.00e4·11-s + 2.23e3i·12-s − 1.63e3·13-s + 1.27e4·14-s + 1.35e4·15-s + 1.88e4·16-s + 2.59e4·17-s + ⋯
L(s)  = 1  − 0.509i·2-s + 0.145i·3-s + 0.740·4-s − 1.84i·5-s + 0.0741·6-s + 0.652i·7-s − 0.886i·8-s + 0.978·9-s − 0.940·10-s − 0.686·11-s + 0.107i·12-s − 0.0572·13-s + 0.332·14-s + 0.268·15-s + 0.288·16-s + 0.310·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.584 + 0.811i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.584 + 0.811i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.972190 - 1.89896i\)
\(L(\frac12)\)  \(\approx\)  \(0.972190 - 1.89896i\)
\(L(5)\)   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 \( 1 + (1.99e6 - 2.77e6i)T \)
good2 \( 1 + 8.15iT - 256T^{2} \)
3 \( 1 - 11.7iT - 6.56e3T^{2} \)
5 \( 1 + 1.15e3iT - 3.90e5T^{2} \)
7 \( 1 - 1.56e3iT - 5.76e6T^{2} \)
11 \( 1 + 1.00e4T + 2.14e8T^{2} \)
13 \( 1 + 1.63e3T + 8.15e8T^{2} \)
17 \( 1 - 2.59e4T + 6.97e9T^{2} \)
19 \( 1 + 1.67e5iT - 1.69e10T^{2} \)
23 \( 1 + 2.25e5T + 7.83e10T^{2} \)
29 \( 1 + 8.29e5iT - 5.00e11T^{2} \)
31 \( 1 + 8.03e5T + 8.52e11T^{2} \)
37 \( 1 + 2.67e5iT - 3.51e12T^{2} \)
41 \( 1 - 2.02e6T + 7.98e12T^{2} \)
47 \( 1 - 6.67e6T + 2.38e13T^{2} \)
53 \( 1 - 1.35e7T + 6.22e13T^{2} \)
59 \( 1 + 1.38e7T + 1.46e14T^{2} \)
61 \( 1 - 2.78e6iT - 1.91e14T^{2} \)
67 \( 1 + 2.07e6T + 4.06e14T^{2} \)
71 \( 1 - 3.64e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.21e7iT - 8.06e14T^{2} \)
79 \( 1 + 2.21e7T + 1.51e15T^{2} \)
83 \( 1 - 1.20e7T + 2.25e15T^{2} \)
89 \( 1 + 4.53e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.49e8T + 7.83e15T^{2} \)
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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

−13.24026514677935958993340182385, −12.56845506812636978014544448729, −11.64814129423278266965610106205, −10.08802872482619592403450819190, −9.012052751770682896553351957597, −7.58195208792194325836416681439, −5.65368385292609410827341514276, −4.29246333975025535904929526341, −2.17711096856830047305012413966, −0.796167258052677394494734866817, 2.02721087124314925517993233463, 3.54173885904885368592389439794, 5.95490803053160597114769898111, 7.15426180277577005700895082028, 7.63418544598004921177890186630, 10.30261889608318417012157450632, 10.66807947448385134809183866166, 12.14928661713604717806557029633, 13.84058686291442840150452439438, 14.72259828748848250972529832571

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