Properties

Degree 2
Conductor 43
Sign $0.381 - 0.924i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 31.0i·2-s − 76.9i·3-s − 710.·4-s + 309. i·5-s + 2.39e3·6-s − 1.60e3i·7-s − 1.41e4i·8-s + 638.·9-s − 9.61e3·10-s − 7.35e3·11-s + 5.47e4i·12-s + 4.21e4·13-s + 4.98e4·14-s + 2.37e4·15-s + 2.57e5·16-s − 5.70e4·17-s + ⋯
L(s)  = 1  + 1.94i·2-s − 0.950i·3-s − 2.77·4-s + 0.494i·5-s + 1.84·6-s − 0.668i·7-s − 3.45i·8-s + 0.0972·9-s − 0.961·10-s − 0.502·11-s + 2.63i·12-s + 1.47·13-s + 1.29·14-s + 0.469·15-s + 3.93·16-s − 0.683·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.381 - 0.924i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.381 - 0.924i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(1.26735 + 0.848317i\)
\(L(\frac12)\)  \(\approx\)  \(1.26735 + 0.848317i\)
\(L(5)\)   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 \( 1 + (-1.30e6 + 3.16e6i)T \)
good2 \( 1 - 31.0iT - 256T^{2} \)
3 \( 1 + 76.9iT - 6.56e3T^{2} \)
5 \( 1 - 309. iT - 3.90e5T^{2} \)
7 \( 1 + 1.60e3iT - 5.76e6T^{2} \)
11 \( 1 + 7.35e3T + 2.14e8T^{2} \)
13 \( 1 - 4.21e4T + 8.15e8T^{2} \)
17 \( 1 + 5.70e4T + 6.97e9T^{2} \)
19 \( 1 - 1.27e5iT - 1.69e10T^{2} \)
23 \( 1 - 2.87e5T + 7.83e10T^{2} \)
29 \( 1 + 2.92e5iT - 5.00e11T^{2} \)
31 \( 1 + 4.78e5T + 8.52e11T^{2} \)
37 \( 1 + 1.22e6iT - 3.51e12T^{2} \)
41 \( 1 - 5.52e6T + 7.98e12T^{2} \)
47 \( 1 + 1.51e6T + 2.38e13T^{2} \)
53 \( 1 - 7.83e6T + 6.22e13T^{2} \)
59 \( 1 - 1.53e7T + 1.46e14T^{2} \)
61 \( 1 + 3.41e6iT - 1.91e14T^{2} \)
67 \( 1 + 3.26e7T + 4.06e14T^{2} \)
71 \( 1 - 4.27e7iT - 6.45e14T^{2} \)
73 \( 1 + 5.32e7iT - 8.06e14T^{2} \)
79 \( 1 - 4.22e7T + 1.51e15T^{2} \)
83 \( 1 + 4.72e7T + 2.25e15T^{2} \)
89 \( 1 + 7.79e7iT - 3.93e15T^{2} \)
97 \( 1 - 5.35e7T + 7.83e15T^{2} \)
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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

−14.47814159148047133146675356872, −13.49887262655069198005964061833, −12.88575540798253069399539766778, −10.55883895699017714486874631367, −8.839487118170347629867327935261, −7.65050576563317165086329279330, −6.88889230928686897369316275236, −5.85786490846415572029457546668, −4.04564924177263132108824199850, −0.813803040489193088465429346494, 1.06562537123398876859868930986, 2.79416524137440292235255615825, 4.17227401587825534740563461316, 5.21087371373387757136099802142, 8.738066886685802819212988304882, 9.250894056925344543320244518652, 10.65889788801733990916489414497, 11.24921701291759686572748862708, 12.73076830729056503640614780788, 13.36824650252867738291705700179

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