Properties

Degree 2
Conductor 43
Sign $0.942 + 0.335i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.65i·2-s − 4.18i·3-s + 2.64·4-s − 45.6i·5-s + 15.3·6-s − 34.3i·7-s + 68.1i·8-s + 63.4·9-s + 166.·10-s − 103.·11-s − 11.0i·12-s + 134.·13-s + 125.·14-s − 191.·15-s − 206.·16-s + 240.·17-s + ⋯
L(s)  = 1  + 0.913i·2-s − 0.465i·3-s + 0.165·4-s − 1.82i·5-s + 0.425·6-s − 0.700i·7-s + 1.06i·8-s + 0.783·9-s + 1.66·10-s − 0.852·11-s − 0.0768i·12-s + 0.798·13-s + 0.640·14-s − 0.850·15-s − 0.807·16-s + 0.831·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.942 + 0.335i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ 0.942 + 0.335i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.64771 - 0.284405i\)
\(L(\frac12)\)  \(\approx\)  \(1.64771 - 0.284405i\)
\(L(3)\)   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 \( 1 + (-1.74e3 - 619. i)T \)
good2 \( 1 - 3.65iT - 16T^{2} \)
3 \( 1 + 4.18iT - 81T^{2} \)
5 \( 1 + 45.6iT - 625T^{2} \)
7 \( 1 + 34.3iT - 2.40e3T^{2} \)
11 \( 1 + 103.T + 1.46e4T^{2} \)
13 \( 1 - 134.T + 2.85e4T^{2} \)
17 \( 1 - 240.T + 8.35e4T^{2} \)
19 \( 1 - 100. iT - 1.30e5T^{2} \)
23 \( 1 + 475.T + 2.79e5T^{2} \)
29 \( 1 - 159. iT - 7.07e5T^{2} \)
31 \( 1 - 1.11e3T + 9.23e5T^{2} \)
37 \( 1 - 2.45e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.06e3T + 2.82e6T^{2} \)
47 \( 1 + 2.93e3T + 4.87e6T^{2} \)
53 \( 1 + 825.T + 7.89e6T^{2} \)
59 \( 1 + 1.24e3T + 1.21e7T^{2} \)
61 \( 1 - 6.20e3iT - 1.38e7T^{2} \)
67 \( 1 - 265.T + 2.01e7T^{2} \)
71 \( 1 - 3.86e3iT - 2.54e7T^{2} \)
73 \( 1 + 8.27e3iT - 2.83e7T^{2} \)
79 \( 1 - 4.95e3T + 3.89e7T^{2} \)
83 \( 1 + 1.35e4T + 4.74e7T^{2} \)
89 \( 1 + 5.35e3iT - 6.27e7T^{2} \)
97 \( 1 - 3.19e3T + 8.85e7T^{2} \)
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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

−15.61740967388467179459445621548, −13.83556209742295413741226771064, −12.98884801233211081220826619687, −11.89720712279267285681024875287, −10.09617295398892639322842721360, −8.360574355937320611138103104276, −7.64403608145323177224283317552, −5.99834149645641786465619607785, −4.60770522893687788492882239185, −1.26527506817634137306877896819, 2.39272410498991029173531096994, 3.63133345149936214909234479107, 6.16465808234016729311250674115, 7.52763847908721756140282321257, 9.788239382902760664303672127467, 10.54943014213267766268435913040, 11.35454833019285171933847209798, 12.65948796811111089698574410887, 14.14659854152982105033833912709, 15.48321337810202890363361235680

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