Properties

Degree 2
Conductor 43
Sign $0.848 - 0.529i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.7i·2-s − 14.6i·3-s − 125.·4-s + 125. i·5-s − 201.·6-s + 485. i·7-s + 839. i·8-s + 513.·9-s + 1.72e3·10-s − 2.19e3·11-s + 1.83e3i·12-s − 2.48e3·13-s + 6.67e3·14-s + 1.84e3·15-s + 3.54e3·16-s − 3.21e3·17-s + ⋯
L(s)  = 1  − 1.71i·2-s − 0.544i·3-s − 1.95·4-s + 1.00i·5-s − 0.935·6-s + 1.41i·7-s + 1.64i·8-s + 0.703·9-s + 1.72·10-s − 1.64·11-s + 1.06i·12-s − 1.13·13-s + 2.43·14-s + 0.546·15-s + 0.864·16-s − 0.654·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.848 - 0.529i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.848 - 0.529i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.524957 + 0.150246i\)
\(L(\frac12)\)  \(\approx\)  \(0.524957 + 0.150246i\)
\(L(4)\)   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 \( 1 + (6.74e4 - 4.20e4i)T \)
good2 \( 1 + 13.7iT - 64T^{2} \)
3 \( 1 + 14.6iT - 729T^{2} \)
5 \( 1 - 125. iT - 1.56e4T^{2} \)
7 \( 1 - 485. iT - 1.17e5T^{2} \)
11 \( 1 + 2.19e3T + 1.77e6T^{2} \)
13 \( 1 + 2.48e3T + 4.82e6T^{2} \)
17 \( 1 + 3.21e3T + 2.41e7T^{2} \)
19 \( 1 - 828. iT - 4.70e7T^{2} \)
23 \( 1 - 1.19e4T + 1.48e8T^{2} \)
29 \( 1 - 3.96e4iT - 5.94e8T^{2} \)
31 \( 1 + 4.16e4T + 8.87e8T^{2} \)
37 \( 1 + 6.79e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.56e4T + 4.75e9T^{2} \)
47 \( 1 - 9.73e4T + 1.07e10T^{2} \)
53 \( 1 + 7.42e4T + 2.21e10T^{2} \)
59 \( 1 + 1.59e5T + 4.21e10T^{2} \)
61 \( 1 + 4.54e4iT - 5.15e10T^{2} \)
67 \( 1 - 5.92e4T + 9.04e10T^{2} \)
71 \( 1 - 4.50e5iT - 1.28e11T^{2} \)
73 \( 1 - 1.02e5iT - 1.51e11T^{2} \)
79 \( 1 - 2.66e3T + 2.43e11T^{2} \)
83 \( 1 + 6.57e5T + 3.26e11T^{2} \)
89 \( 1 + 6.80e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.25e6T + 8.32e11T^{2} \)
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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

−14.52005158594166903252852193317, −12.88065514214944358866107693393, −12.57652174809075120326791372844, −11.17340691049427103544929814151, −10.31613290066962044645883177924, −9.036788419776929103467556561617, −7.26250440507474435129563692347, −5.12045775080926097221055994764, −2.90206634267096550531962810278, −2.10949493294047981989542394032, 0.24753120798455589039387095891, 4.45310790883934021642346816437, 5.10686573811960652581143148036, 7.08585163140240941954703200003, 7.911872534636363782375579708783, 9.370688642298319301524758420436, 10.50232920770636812902650048084, 12.95748587527676232023790871643, 13.52800810741256357094594772999, 15.04283944237160121293697243920

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