Properties

Degree 2
Conductor 43
Sign $0.820 + 0.572i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.15i·2-s + (41.5 + 24.0i)3-s − 2.46·4-s + (−26.4 − 15.2i)5-s + (195. − 338. i)6-s + (6.27 − 3.62i)7-s − 501. i·8-s + (787. + 1.36e3i)9-s + (−124. + 215. i)10-s + 1.43e3·11-s + (−102. − 59.0i)12-s + (1.14e3 + 1.98e3i)13-s + (−29.5 − 51.1i)14-s + (−733. − 1.27e3i)15-s − 4.24e3·16-s + (−3.89e3 − 6.75e3i)17-s + ⋯
L(s)  = 1  − 1.01i·2-s + (1.53 + 0.889i)3-s − 0.0384·4-s + (−0.211 − 0.122i)5-s + (0.905 − 1.56i)6-s + (0.0182 − 0.0105i)7-s − 0.979i·8-s + (1.08 + 1.87i)9-s + (−0.124 + 0.215i)10-s + 1.08·11-s + (−0.0592 − 0.0341i)12-s + (0.521 + 0.903i)13-s + (−0.0107 − 0.0186i)14-s + (−0.217 − 0.376i)15-s − 1.03·16-s + (−0.793 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.572i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.820 + 0.572i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.820 + 0.572i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(2.93222 - 0.921630i\)
\(L(\frac12)\)  \(\approx\)  \(2.93222 - 0.921630i\)
\(L(4)\)   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 \( 1 + (-3.97e4 + 6.88e4i)T \)
good2 \( 1 + 8.15iT - 64T^{2} \)
3 \( 1 + (-41.5 - 24.0i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (26.4 + 15.2i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (-6.27 + 3.62i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 - 1.43e3T + 1.77e6T^{2} \)
13 \( 1 + (-1.14e3 - 1.98e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (3.89e3 + 6.75e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-5.66e3 - 3.26e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-1.00e3 + 1.73e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (3.37e4 - 1.95e4i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (1.14e4 - 1.97e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (3.02e4 + 1.74e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 3.35e4T + 4.75e9T^{2} \)
47 \( 1 - 1.04e4T + 1.07e10T^{2} \)
53 \( 1 + (1.55e4 - 2.69e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + 1.40e5T + 4.21e10T^{2} \)
61 \( 1 + (1.17e5 - 6.79e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.15e5 - 1.99e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (-3.91e5 + 2.25e5i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (3.74e5 - 2.16e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-7.84e4 - 1.35e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-3.72e5 + 6.45e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-9.77e4 - 5.64e4i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 1.45e6T + 8.32e11T^{2} \)
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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

−14.37049910171479655813400726135, −13.61131717909125580039249707162, −12.01927876764148997478632027900, −10.88873301254496757315493277755, −9.517747530290514621980972011365, −8.951433150700091248232278675067, −7.16499823722996097463433555518, −4.27049556702719646671387556547, −3.29081816117040421943474768927, −1.80861443523357726933839457634, 1.78548617528853638185254474189, 3.53563441523668036893925264963, 6.17221422966472795848293111401, 7.33868989017473884765671475256, 8.193584972772834083513246685334, 9.224967731988858013674769177538, 11.37779571314582465659255642796, 12.96829222272331363296450291980, 13.87268687791888537133395185380, 15.03659360893214917938528685111

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