Properties

Degree 2
Conductor 43
Sign $0.999 - 0.0299i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.80 − 7.28i)2-s + (−18.0 − 2.72i)3-s + (−12.1 + 53.3i)4-s + (−87.0 − 59.3i)5-s + (85.1 + 147. i)6-s + (−25.6 + 44.5i)7-s + (190. − 91.9i)8-s + (86.9 + 26.8i)9-s + (73.2 + 977. i)10-s + (−89.3 − 391. i)11-s + (365. − 931. i)12-s + (66.5 − 887. i)13-s + (473. − 71.3i)14-s + (1.41e3 + 1.30e3i)15-s + (−199. − 96.0i)16-s + (938. − 639. i)17-s + ⋯
L(s)  = 1  + (−1.02 − 1.28i)2-s + (−1.15 − 0.174i)3-s + (−0.380 + 1.66i)4-s + (−1.55 − 1.06i)5-s + (0.965 + 1.67i)6-s + (−0.198 + 0.343i)7-s + (1.05 − 0.507i)8-s + (0.357 + 0.110i)9-s + (0.231 + 3.09i)10-s + (−0.222 − 0.975i)11-s + (0.732 − 1.86i)12-s + (0.109 − 1.45i)13-s + (0.645 − 0.0972i)14-s + (1.61 + 1.50i)15-s + (−0.194 − 0.0937i)16-s + (0.787 − 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0299i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.999 - 0.0299i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 0.999 - 0.0299i)\)
\(L(3)\)  \(\approx\)  \(0.0108519 + 0.000162340i\)
\(L(\frac12)\)  \(\approx\)  \(0.0108519 + 0.000162340i\)
\(L(\frac{7}{2})\)   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 \( 1 + (1.19e3 + 1.20e4i)T \)
good2 \( 1 + (5.80 + 7.28i)T + (-7.12 + 31.1i)T^{2} \)
3 \( 1 + (18.0 + 2.72i)T + (232. + 71.6i)T^{2} \)
5 \( 1 + (87.0 + 59.3i)T + (1.14e3 + 2.90e3i)T^{2} \)
7 \( 1 + (25.6 - 44.5i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (89.3 + 391. i)T + (-1.45e5 + 6.98e4i)T^{2} \)
13 \( 1 + (-66.5 + 887. i)T + (-3.67e5 - 5.53e4i)T^{2} \)
17 \( 1 + (-938. + 639. i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (2.47e3 - 762. i)T + (2.04e6 - 1.39e6i)T^{2} \)
23 \( 1 + (785. - 728. i)T + (4.80e5 - 6.41e6i)T^{2} \)
29 \( 1 + (1.43e3 - 216. i)T + (1.95e7 - 6.04e6i)T^{2} \)
31 \( 1 + (16.1 - 41.0i)T + (-2.09e7 - 1.94e7i)T^{2} \)
37 \( 1 + (-125. - 216. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + (8.96e3 + 1.12e4i)T + (-2.57e7 + 1.12e8i)T^{2} \)
47 \( 1 + (939. - 4.11e3i)T + (-2.06e8 - 9.95e7i)T^{2} \)
53 \( 1 + (912. + 1.21e4i)T + (-4.13e8 + 6.23e7i)T^{2} \)
59 \( 1 + (-1.09e4 - 5.28e3i)T + (4.45e8 + 5.58e8i)T^{2} \)
61 \( 1 + (1.07e4 + 2.72e4i)T + (-6.19e8 + 5.74e8i)T^{2} \)
67 \( 1 + (-1.44e4 + 4.46e3i)T + (1.11e9 - 7.60e8i)T^{2} \)
71 \( 1 + (-2.04e4 - 1.90e4i)T + (1.34e8 + 1.79e9i)T^{2} \)
73 \( 1 + (-2.37e3 + 3.17e4i)T + (-2.04e9 - 3.08e8i)T^{2} \)
79 \( 1 + (1.17e4 - 2.02e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-6.24e4 - 9.40e3i)T + (3.76e9 + 1.16e9i)T^{2} \)
89 \( 1 + (1.17e5 + 1.77e4i)T + (5.33e9 + 1.64e9i)T^{2} \)
97 \( 1 + (-7.03e3 - 3.08e4i)T + (-7.73e9 + 3.72e9i)T^{2} \)
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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

−12.74878141770319016951110007153, −12.24316983334325430852279694652, −11.36432567660638746755885229542, −10.51296012586256294566680271891, −8.729611020927784509751855155749, −7.948687958222323335624116152750, −5.48953264139694944680688544171, −3.50118857099269772920689444585, −0.73151729601478495408530428294, −0.01606724505877742011818161944, 4.37356348587824790878500917020, 6.41923048417596985313795525667, 7.07248672882299621608100369992, 8.296119426323859042639121313582, 10.10594481554595715072249493164, 11.07508047897098407293430853412, 12.16740624703020191649819109455, 14.58771250070550149312552151916, 15.25273485985669900158174309615, 16.34692795381927283242476610717

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