Properties

Degree 2
Conductor 43
Sign $0.762 + 0.646i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.33i·2-s + 16.8i·3-s − 5.39·4-s − 47.9i·5-s + 139.·6-s + 493. i·7-s − 488. i·8-s + 446.·9-s − 399.·10-s + 2.01e3·11-s − 90.6i·12-s − 124.·13-s + 4.11e3·14-s + 804.·15-s − 4.41e3·16-s + 3.85e3·17-s + ⋯
L(s)  = 1  − 1.04i·2-s + 0.622i·3-s − 0.0843·4-s − 0.383i·5-s + 0.647·6-s + 1.43i·7-s − 0.953i·8-s + 0.612·9-s − 0.399·10-s + 1.51·11-s − 0.0524i·12-s − 0.0564·13-s + 1.49·14-s + 0.238·15-s − 1.07·16-s + 0.785·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.762 + 0.646i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.762 + 0.646i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.99206 - 0.730459i\)
\(L(\frac12)\)  \(\approx\)  \(1.99206 - 0.730459i\)
\(L(4)\)   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 \( 1 + (6.06e4 + 5.13e4i)T \)
good2 \( 1 + 8.33iT - 64T^{2} \)
3 \( 1 - 16.8iT - 729T^{2} \)
5 \( 1 + 47.9iT - 1.56e4T^{2} \)
7 \( 1 - 493. iT - 1.17e5T^{2} \)
11 \( 1 - 2.01e3T + 1.77e6T^{2} \)
13 \( 1 + 124.T + 4.82e6T^{2} \)
17 \( 1 - 3.85e3T + 2.41e7T^{2} \)
19 \( 1 + 1.11e4iT - 4.70e7T^{2} \)
23 \( 1 + 1.13e3T + 1.48e8T^{2} \)
29 \( 1 - 2.43e4iT - 5.94e8T^{2} \)
31 \( 1 + 1.24e4T + 8.87e8T^{2} \)
37 \( 1 - 8.46e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.25e5T + 4.75e9T^{2} \)
47 \( 1 + 2.00e5T + 1.07e10T^{2} \)
53 \( 1 + 1.24e5T + 2.21e10T^{2} \)
59 \( 1 + 1.23e5T + 4.21e10T^{2} \)
61 \( 1 + 2.87e4iT - 5.15e10T^{2} \)
67 \( 1 - 2.89e5T + 9.04e10T^{2} \)
71 \( 1 + 4.20e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.30e5iT - 1.51e11T^{2} \)
79 \( 1 + 6.47e5T + 2.43e11T^{2} \)
83 \( 1 + 7.67e4T + 3.26e11T^{2} \)
89 \( 1 + 1.69e4iT - 4.96e11T^{2} \)
97 \( 1 + 1.10e6T + 8.32e11T^{2} \)
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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

−14.73258861669851868094269376498, −12.88353478584526098059814595501, −12.06051062137669702178477943944, −11.11907996775533611028894900365, −9.639994611931490735103780988506, −8.973808739410430770920585706274, −6.66614791708295428360474149006, −4.78082878143752584815840741593, −3.16874342797266245680335170351, −1.42952992914123833648503340897, 1.36777951588150601660165496415, 4.01517503908801916029854601221, 6.19645310091569722254658749569, 7.10980877275988241921691790350, 7.909402194382690945283778232824, 9.880841114045650211163919849455, 11.26251484707128939586889924151, 12.65035655579655338221007404882, 14.21611385039064574166039860668, 14.48979949381818488432297805704

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