Properties

Degree 2
Conductor 43
Sign $0.278 + 0.960i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.45 + 5.10i)2-s + (−50.4 − 3.78i)3-s + (19.8 + 24.9i)4-s + (−59.6 + 193. i)5-s + (143. − 248. i)6-s + (−466. + 269. i)7-s + (−529. + 120. i)8-s + (1.81e3 + 273. i)9-s + (−840. − 780. i)10-s + (−34.7 + 43.5i)11-s + (−908. − 1.33e3i)12-s + (261. − 242. i)13-s + (−227. − 3.04e3i)14-s + (3.74e3 − 9.53e3i)15-s + (231. − 1.01e3i)16-s + (769. − 237. i)17-s + ⋯
L(s)  = 1  + (−0.307 + 0.638i)2-s + (−1.86 − 0.140i)3-s + (0.310 + 0.389i)4-s + (−0.477 + 1.54i)5-s + (0.664 − 1.15i)6-s + (−1.35 + 0.784i)7-s + (−1.03 + 0.236i)8-s + (2.48 + 0.374i)9-s + (−0.840 − 0.780i)10-s + (−0.0260 + 0.0326i)11-s + (−0.525 − 0.771i)12-s + (0.118 − 0.110i)13-s + (−0.0830 − 1.10i)14-s + (1.10 − 2.82i)15-s + (0.0565 − 0.247i)16-s + (0.156 − 0.0483i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.278 + 0.960i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.278 + 0.960i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.240625 - 0.180856i\)
\(L(\frac12)\)  \(\approx\)  \(0.240625 - 0.180856i\)
\(L(4)\)   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 \( 1 + (6.78e4 - 4.14e4i)T \)
good2 \( 1 + (2.45 - 5.10i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (50.4 + 3.78i)T + (720. + 108. i)T^{2} \)
5 \( 1 + (59.6 - 193. i)T + (-1.29e4 - 8.80e3i)T^{2} \)
7 \( 1 + (466. - 269. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (34.7 - 43.5i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (-261. + 242. i)T + (3.60e5 - 4.81e6i)T^{2} \)
17 \( 1 + (-769. + 237. i)T + (1.99e7 - 1.35e7i)T^{2} \)
19 \( 1 + (-396. - 2.62e3i)T + (-4.49e7 + 1.38e7i)T^{2} \)
23 \( 1 + (-6.63e3 - 1.69e4i)T + (-1.08e8 + 1.00e8i)T^{2} \)
29 \( 1 + (-1.78e4 + 1.33e3i)T + (5.88e8 - 8.86e7i)T^{2} \)
31 \( 1 + (3.12e4 - 2.12e4i)T + (3.24e8 - 8.26e8i)T^{2} \)
37 \( 1 + (-6.32e4 - 3.65e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (-6.12e4 - 2.95e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (-1.31e4 - 1.64e4i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (-1.02e4 - 9.47e3i)T + (1.65e9 + 2.21e10i)T^{2} \)
59 \( 1 + (-2.58e4 + 1.13e5i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (2.40e3 - 3.52e3i)T + (-1.88e10 - 4.79e10i)T^{2} \)
67 \( 1 + (1.95e5 - 2.94e4i)T + (8.64e10 - 2.66e10i)T^{2} \)
71 \( 1 + (-3.09e5 - 1.21e5i)T + (9.39e10 + 8.71e10i)T^{2} \)
73 \( 1 + (2.67e5 + 2.88e5i)T + (-1.13e10 + 1.50e11i)T^{2} \)
79 \( 1 + (2.80e4 + 4.86e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-4.99e4 + 6.66e5i)T + (-3.23e11 - 4.87e10i)T^{2} \)
89 \( 1 + (9.02e5 + 6.76e4i)T + (4.91e11 + 7.40e10i)T^{2} \)
97 \( 1 + (1.12e6 - 1.41e6i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−15.86910098137893727060494852867, −15.11022595294420619176746566846, −12.84748517903810039808400973669, −11.85887754207011271151144985952, −11.08429668704273293426641922969, −9.797425581071259717013286024347, −7.42701985705904002294235717092, −6.56877370237766303211676583836, −5.85991404957838549821895981357, −3.20268625710728028604560515151, 0.28236898470802542627387656290, 0.863808196805134014911839271038, 4.26116484074625831803685621762, 5.69216786973031986397948743504, 6.86254573368411891746318575466, 9.306111919192273788338469168752, 10.32827755959679296240678773749, 11.32077245779546924831375619616, 12.43792121837843491299366455730, 12.91235854089164682309596077418

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