Properties

Degree 2
Conductor 43
Sign $-0.572 + 0.819i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.92 − 8.14i)2-s + (7.09 + 0.532i)3-s + (−11.0 − 13.8i)4-s + (43.9 − 142. i)5-s + (32.1 − 55.7i)6-s + (−72.9 + 42.1i)7-s + (407. − 93.0i)8-s + (−670. − 101. i)9-s + (−987. − 916. i)10-s + (1.14e3 − 1.42e3i)11-s + (−71.2 − 104. i)12-s + (798. − 740. i)13-s + (56.9 + 759. i)14-s + (387. − 987. i)15-s + (1.09e3 − 4.79e3i)16-s + (−9.19e3 + 2.83e3i)17-s + ⋯
L(s)  = 1  + (0.490 − 1.01i)2-s + (0.262 + 0.0197i)3-s + (−0.173 − 0.217i)4-s + (0.351 − 1.13i)5-s + (0.149 − 0.258i)6-s + (−0.212 + 0.122i)7-s + (0.796 − 0.181i)8-s + (−0.920 − 0.138i)9-s + (−0.987 − 0.916i)10-s + (0.856 − 1.07i)11-s + (−0.0412 − 0.0604i)12-s + (0.363 − 0.337i)13-s + (0.0207 + 0.276i)14-s + (0.114 − 0.292i)15-s + (0.267 − 1.17i)16-s + (−1.87 + 0.577i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.572 + 0.819i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.572 + 0.819i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1.17058 - 2.24664i\)
\(L(\frac12)\)  \(\approx\)  \(1.17058 - 2.24664i\)
\(L(4)\)   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 \( 1 + (7.48e4 + 2.67e4i)T \)
good2 \( 1 + (-3.92 + 8.14i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (-7.09 - 0.532i)T + (720. + 108. i)T^{2} \)
5 \( 1 + (-43.9 + 142. i)T + (-1.29e4 - 8.80e3i)T^{2} \)
7 \( 1 + (72.9 - 42.1i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (-1.14e3 + 1.42e3i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (-798. + 740. i)T + (3.60e5 - 4.81e6i)T^{2} \)
17 \( 1 + (9.19e3 - 2.83e3i)T + (1.99e7 - 1.35e7i)T^{2} \)
19 \( 1 + (423. + 2.81e3i)T + (-4.49e7 + 1.38e7i)T^{2} \)
23 \( 1 + (-7.07e3 - 1.80e4i)T + (-1.08e8 + 1.00e8i)T^{2} \)
29 \( 1 + (-1.73e4 + 1.30e3i)T + (5.88e8 - 8.86e7i)T^{2} \)
31 \( 1 + (-4.49e4 + 3.06e4i)T + (3.24e8 - 8.26e8i)T^{2} \)
37 \( 1 + (-8.45e4 - 4.87e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (-2.64e3 - 1.27e3i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (-5.36e4 - 6.72e4i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (-1.25e5 - 1.16e5i)T + (1.65e9 + 2.21e10i)T^{2} \)
59 \( 1 + (4.58e4 - 2.01e5i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (-2.41e4 + 3.54e4i)T + (-1.88e10 - 4.79e10i)T^{2} \)
67 \( 1 + (-1.19e5 + 1.80e4i)T + (8.64e10 - 2.66e10i)T^{2} \)
71 \( 1 + (4.96e5 + 1.94e5i)T + (9.39e10 + 8.71e10i)T^{2} \)
73 \( 1 + (3.70e5 + 3.99e5i)T + (-1.13e10 + 1.50e11i)T^{2} \)
79 \( 1 + (-2.49e4 - 4.31e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (6.06e4 - 8.08e5i)T + (-3.23e11 - 4.87e10i)T^{2} \)
89 \( 1 + (2.56e5 + 1.91e4i)T + (4.91e11 + 7.40e10i)T^{2} \)
97 \( 1 + (1.37e5 - 1.72e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−13.54110818473312551786344826565, −13.32819069647878875869301469734, −11.83612843876109686839929611742, −11.08823917812096542920332776503, −9.312296967395234051475310681791, −8.398556553060604183820765245219, −6.08616901233166624392216079246, −4.38815099214087826092630180560, −2.86557991416073943702624113481, −1.08101320324286086479859431454, 2.42039312674184992126291178368, 4.52505173380167509546563609049, 6.43369171720076115134687435504, 6.87010943945563629618405453764, 8.646609346395234931336019464062, 10.29157476149715988272174422462, 11.44850560046813991256314545375, 13.31187916587623431248082769782, 14.38246482068957037279987110352, 14.75524265157364872841079272637

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