Properties

Degree 2
Conductor 43
Sign $0.985 + 0.171i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.5 − 0.866i)3-s − 4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−1.5 + 2.59i)7-s − 3·8-s + (1 − 1.73i)9-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)12-s + (2.5 − 4.33i)13-s + (−1.5 + 2.59i)14-s + (0.499 − 0.866i)15-s − 16-s + (−1.5 + 2.59i)17-s + (1 − 1.73i)18-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.288 − 0.499i)3-s − 0.5·4-s + (0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (−0.566 + 0.981i)7-s − 1.06·8-s + (0.333 − 0.577i)9-s + (0.158 + 0.273i)10-s + (0.144 + 0.249i)12-s + (0.693 − 1.20i)13-s + (−0.400 + 0.694i)14-s + (0.129 − 0.223i)15-s − 0.250·16-s + (−0.363 + 0.630i)17-s + (0.235 − 0.408i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.985 + 0.171i$
motivic weight  =  \(1\)
character  :  $\chi_{43} (36, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :1/2),\ 0.985 + 0.171i)\)
\(L(1)\)  \(\approx\)  \(0.880980 - 0.0761557i\)
\(L(\frac12)\)  \(\approx\)  \(0.880980 - 0.0761557i\)
\(L(\frac{3}{2})\)   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 \( 1 + (4 + 5.19i)T \)
good2 \( 1 - T + 2T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−15.48698718155727381100159608334, −14.93259322616907854553657847490, −13.25989587995379888651411950046, −12.83759326146348050290394206512, −11.62116892256875750743888082634, −9.864497437365040705722664129784, −8.577406316803923368617198473940, −6.52279276308513502587000590701, −5.51855702707420985771365031225, −3.35866792301398925150096351414, 3.95105490868582339229663793418, 5.03108871220078792971238414479, 6.76486336296373504490087508573, 8.848449434936333430085111719598, 10.00079602267096372848508341429, 11.32610889315071847199510580771, 12.98060071194356131373890504545, 13.52877102884076407184858132043, 14.68264660679604815517282447134, 16.25893957648634954297268744775

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