Properties

Degree 2
Conductor 43
Sign $0.630 - 0.776i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.74 − 7.78i)2-s + (23.1 − 48.0i)3-s + (113. − 141. i)4-s + (−1.13e3 − 259. i)5-s − 460.·6-s + 3.46e3i·7-s + (−3.68e3 − 840. i)8-s + (2.31e3 + 2.90e3i)9-s + (2.23e3 + 9.80e3i)10-s + (3.83e3 + 4.80e3i)11-s + (−4.19e3 − 8.71e3i)12-s + (−115. + 507. i)13-s + (2.69e4 − 1.29e4i)14-s + (−3.87e4 + 4.85e4i)15-s + (−3.07e3 − 1.34e4i)16-s + (−1.58e4 − 6.96e4i)17-s + ⋯
L(s)  = 1  + (−0.234 − 0.486i)2-s + (0.285 − 0.593i)3-s + (0.441 − 0.554i)4-s + (−1.81 − 0.414i)5-s − 0.355·6-s + 1.44i·7-s + (−0.899 − 0.205i)8-s + (0.353 + 0.443i)9-s + (0.223 + 0.980i)10-s + (0.261 + 0.328i)11-s + (−0.202 − 0.420i)12-s + (−0.00405 + 0.0177i)13-s + (0.702 − 0.338i)14-s + (−0.764 + 0.958i)15-s + (−0.0469 − 0.205i)16-s + (−0.190 − 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.630 - 0.776i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.630 - 0.776i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.622711 + 0.296564i\)
\(L(\frac12)\)  \(\approx\)  \(0.622711 + 0.296564i\)
\(L(5)\)   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 \( 1 + (5.24e5 - 3.37e6i)T \)
good2 \( 1 + (3.74 + 7.78i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (-23.1 + 48.0i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (1.13e3 + 259. i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 - 3.46e3iT - 5.76e6T^{2} \)
11 \( 1 + (-3.83e3 - 4.80e3i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (115. - 507. i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (1.58e4 + 6.96e4i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (-4.17e4 - 3.33e4i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (-5.70e4 - 7.15e4i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (-1.11e4 - 2.31e4i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (1.53e6 - 7.40e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 - 2.51e6iT - 3.51e12T^{2} \)
41 \( 1 + (1.47e6 - 7.10e5i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (3.16e6 - 3.96e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (-1.11e6 - 4.88e6i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (4.67e6 + 2.04e7i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (6.02e6 - 1.25e7i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (-1.38e7 + 1.73e7i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (-2.99e7 - 2.38e7i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (7.01e6 + 1.60e6i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 + 4.74e7T + 1.51e15T^{2} \)
83 \( 1 + (3.10e7 + 1.49e7i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (2.33e7 - 4.85e7i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (-1.09e8 - 1.37e8i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−14.64333611432635840681013363963, −12.67544794029265344662430985190, −11.95677632833709364413590588888, −11.16357387991859766880236926503, −9.332990442708727183571135683140, −8.163183954568981040979173240439, −6.94667123920799555256512412339, −5.01866242370516979918003880846, −2.99145958669976239639215885427, −1.44076165173520381432732830796, 0.28370854153701518538833498806, 3.54751477057883523512459600318, 3.99317859380041626908866337801, 6.88602923451375069132634416315, 7.55500961197697428714151383949, 8.741650027243694281799206201568, 10.58792423847000689832067994515, 11.47853447393006493413166912329, 12.72241843357380535610400516691, 14.56892784345685006995085114066

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