Properties

Degree 2
Conductor 43
Sign $0.318 - 0.947i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.22 + 2.79i)2-s + (1.24 + 0.187i)3-s + (−1.06 + 4.64i)4-s + (6.85 + 4.67i)5-s + (2.24 + 3.88i)6-s + (−6.94 + 12.0i)7-s + (10.4 − 5.01i)8-s + (−24.2 − 7.49i)9-s + (2.21 + 29.5i)10-s + (−7.71 − 33.7i)11-s + (−2.18 + 5.56i)12-s + (2.06 − 27.6i)13-s + (−49.0 + 7.39i)14-s + (7.63 + 7.08i)15-s + (71.5 + 34.4i)16-s + (35.8 − 24.4i)17-s + ⋯
L(s)  = 1  + (0.787 + 0.987i)2-s + (0.238 + 0.0359i)3-s + (−0.132 + 0.580i)4-s + (0.613 + 0.417i)5-s + (0.152 + 0.264i)6-s + (−0.375 + 0.649i)7-s + (0.460 − 0.221i)8-s + (−0.899 − 0.277i)9-s + (0.0700 + 0.934i)10-s + (−0.211 − 0.926i)11-s + (−0.0525 + 0.133i)12-s + (0.0441 − 0.589i)13-s + (−0.937 + 0.141i)14-s + (0.131 + 0.121i)15-s + (1.11 + 0.538i)16-s + (0.511 − 0.348i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.318 - 0.947i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ 0.318 - 0.947i)\)
\(L(2)\)  \(\approx\)  \(1.68350 + 1.21016i\)
\(L(\frac12)\)  \(\approx\)  \(1.68350 + 1.21016i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-272. - 71.2i)T \)
good2 \( 1 + (-2.22 - 2.79i)T + (-1.78 + 7.79i)T^{2} \)
3 \( 1 + (-1.24 - 0.187i)T + (25.8 + 7.95i)T^{2} \)
5 \( 1 + (-6.85 - 4.67i)T + (45.6 + 116. i)T^{2} \)
7 \( 1 + (6.94 - 12.0i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (7.71 + 33.7i)T + (-1.19e3 + 577. i)T^{2} \)
13 \( 1 + (-2.06 + 27.6i)T + (-2.17e3 - 327. i)T^{2} \)
17 \( 1 + (-35.8 + 24.4i)T + (1.79e3 - 4.57e3i)T^{2} \)
19 \( 1 + (-18.2 + 5.64i)T + (5.66e3 - 3.86e3i)T^{2} \)
23 \( 1 + (41.5 - 38.5i)T + (909. - 1.21e4i)T^{2} \)
29 \( 1 + (191. - 28.8i)T + (2.33e4 - 7.18e3i)T^{2} \)
31 \( 1 + (74.7 - 190. i)T + (-2.18e4 - 2.02e4i)T^{2} \)
37 \( 1 + (5.49 + 9.51i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-144. - 180. i)T + (-1.53e4 + 6.71e4i)T^{2} \)
47 \( 1 + (85.7 - 375. i)T + (-9.35e4 - 4.50e4i)T^{2} \)
53 \( 1 + (-0.792 - 10.5i)T + (-1.47e5 + 2.21e4i)T^{2} \)
59 \( 1 + (548. + 264. i)T + (1.28e5 + 1.60e5i)T^{2} \)
61 \( 1 + (131. + 334. i)T + (-1.66e5 + 1.54e5i)T^{2} \)
67 \( 1 + (-267. + 82.5i)T + (2.48e5 - 1.69e5i)T^{2} \)
71 \( 1 + (-405. - 376. i)T + (2.67e4 + 3.56e5i)T^{2} \)
73 \( 1 + (-74.9 + 9.99e2i)T + (-3.84e5 - 5.79e4i)T^{2} \)
79 \( 1 + (430. - 746. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-240. - 36.2i)T + (5.46e5 + 1.68e5i)T^{2} \)
89 \( 1 + (-1.26e3 - 191. i)T + (6.73e5 + 2.07e5i)T^{2} \)
97 \( 1 + (378. + 1.65e3i)T + (-8.22e5 + 3.95e5i)T^{2} \)
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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

−15.54701568058201351225210983802, −14.42499391367166151785439070020, −13.85125179242728852241265703326, −12.59426113817154419856968810930, −10.93196160127730441137084968495, −9.399711072066825746256960402822, −7.88652365980430629385795611416, −6.19109890488790961329999420657, −5.51794304754777184504003376903, −3.15262181150539857913253247259, 2.10457899140966925472759299770, 3.91499839544201542404745254261, 5.51369532246337881426092247780, 7.60767501910792425324436126561, 9.402020964789085883828206883542, 10.60214952159928168399815826626, 11.84953454188535998506429338337, 12.99582652616029805269532270904, 13.71966969455080711560573985163, 14.72843510132713918504541839721

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