Properties

Degree 2
Conductor 43
Sign $0.846 + 0.533i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 50.9i·2-s − 197. i·3-s − 1.57e3·4-s + 5.53e3i·5-s + 1.00e4·6-s − 2.13e4i·7-s − 2.80e4i·8-s + 1.99e4·9-s − 2.81e5·10-s − 1.31e5·11-s + 3.11e5i·12-s − 5.88e5·13-s + 1.09e6·14-s + 1.09e6·15-s − 1.81e5·16-s − 2.70e5·17-s + ⋯
L(s)  = 1  + 1.59i·2-s − 0.814i·3-s − 1.53·4-s + 1.76i·5-s + 1.29·6-s − 1.27i·7-s − 0.857i·8-s + 0.337·9-s − 2.81·10-s − 0.813·11-s + 1.25i·12-s − 1.58·13-s + 2.02·14-s + 1.44·15-s − 0.172·16-s − 0.190·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.846 + 0.533i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ 0.846 + 0.533i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.581972 - 0.168032i\)
\(L(\frac12)\)  \(\approx\)  \(0.581972 - 0.168032i\)
\(L(6)\)   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 \( 1 + (1.24e8 + 7.83e7i)T \)
good2 \( 1 - 50.9iT - 1.02e3T^{2} \)
3 \( 1 + 197. iT - 5.90e4T^{2} \)
5 \( 1 - 5.53e3iT - 9.76e6T^{2} \)
7 \( 1 + 2.13e4iT - 2.82e8T^{2} \)
11 \( 1 + 1.31e5T + 2.59e10T^{2} \)
13 \( 1 + 5.88e5T + 1.37e11T^{2} \)
17 \( 1 + 2.70e5T + 2.01e12T^{2} \)
19 \( 1 + 2.93e6iT - 6.13e12T^{2} \)
23 \( 1 - 1.08e7T + 4.14e13T^{2} \)
29 \( 1 - 7.43e6iT - 4.20e14T^{2} \)
31 \( 1 - 4.62e7T + 8.19e14T^{2} \)
37 \( 1 + 6.79e7iT - 4.80e15T^{2} \)
41 \( 1 - 7.10e7T + 1.34e16T^{2} \)
47 \( 1 + 3.33e8T + 5.25e16T^{2} \)
53 \( 1 + 3.32e8T + 1.74e17T^{2} \)
59 \( 1 + 7.48e7T + 5.11e17T^{2} \)
61 \( 1 + 7.63e8iT - 7.13e17T^{2} \)
67 \( 1 - 5.79e8T + 1.82e18T^{2} \)
71 \( 1 + 9.90e8iT - 3.25e18T^{2} \)
73 \( 1 - 8.61e8iT - 4.29e18T^{2} \)
79 \( 1 + 4.77e9T + 9.46e18T^{2} \)
83 \( 1 + 1.89e9T + 1.55e19T^{2} \)
89 \( 1 + 1.53e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.02e10T + 7.37e19T^{2} \)
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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

−13.95114490380880224851264434384, −13.09419108410607017300997040676, −11.07491384987218884286657457593, −9.920389214639071593147379646586, −7.75026857705843988477712281575, −7.06928887488424587104169870168, −6.71789183884632889947821233085, −4.77247777656190958486061492289, −2.67643417536565340648204979096, −0.20026816111082491605668673573, 1.37764731546882843352867658393, 2.76538519659414784040187339850, 4.57740032168214080314792014591, 5.12692756723572785191655558831, 8.365167381543725122224010116894, 9.455133204143852402268452304012, 10.02944870303087632037833833756, 11.69434757943850099401892282969, 12.50177130459597587401822905072, 13.09980956694168452395320528919

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