Properties

Degree 2
Conductor 43
Sign $-0.739 - 0.673i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.86 − 12.3i)2-s + (−10.7 − 1.61i)3-s + (−27.2 + 119. i)4-s + (180. + 122. i)5-s + (85.6 + 148. i)6-s + (797. − 1.38e3i)7-s + (−79.5 + 38.2i)8-s + (−1.97e3 − 610. i)9-s + (−258. − 3.44e3i)10-s + (−774. − 3.39e3i)11-s + (484. − 1.23e3i)12-s + (−123. + 1.65e3i)13-s + (−2.49e4 + 3.76e3i)14-s + (−1.73e3 − 1.60e3i)15-s + (1.53e4 + 7.40e3i)16-s + (1.62e4 − 1.10e4i)17-s + ⋯
L(s)  = 1  + (−0.872 − 1.09i)2-s + (−0.228 − 0.0345i)3-s + (−0.212 + 0.932i)4-s + (0.645 + 0.439i)5-s + (0.161 + 0.280i)6-s + (0.879 − 1.52i)7-s + (−0.0549 + 0.0264i)8-s + (−0.904 − 0.278i)9-s + (−0.0816 − 1.08i)10-s + (−0.175 − 0.768i)11-s + (0.0809 − 0.206i)12-s + (−0.0156 + 0.208i)13-s + (−2.43 + 0.366i)14-s + (−0.132 − 0.123i)15-s + (0.938 + 0.451i)16-s + (0.802 − 0.546i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.739 - 0.673i$
motivic weight  =  \(7\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ -0.739 - 0.673i)\)
\(L(4)\)  \(\approx\)  \(0.196655 + 0.507886i\)
\(L(\frac12)\)  \(\approx\)  \(0.196655 + 0.507886i\)
\(L(\frac{9}{2})\)   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 \( 1 + (-3.23e5 + 4.08e5i)T \)
good2 \( 1 + (9.86 + 12.3i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (10.7 + 1.61i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (-180. - 122. i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (-797. + 1.38e3i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (774. + 3.39e3i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (123. - 1.65e3i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (-1.62e4 + 1.10e4i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (2.21e4 - 6.81e3i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (5.43e4 - 5.04e4i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (1.92e5 - 2.89e4i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (9.02e4 - 2.30e5i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (1.68e5 + 2.91e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (2.07e5 + 2.60e5i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (1.05e5 - 4.63e5i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (-1.38e5 - 1.85e6i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (-4.11e5 - 1.98e5i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (4.54e5 + 1.15e6i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (2.01e6 - 6.22e5i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (1.14e6 + 1.06e6i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (-4.23e5 + 5.64e6i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (-2.60e6 + 4.51e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (4.87e6 + 7.35e5i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (-1.08e7 - 1.63e6i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (-2.71e5 - 1.18e6i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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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

−13.82936176385351315535769115952, −12.03071213678320251832097367386, −10.91230991128153986218562739799, −10.44647010498998242836144314354, −9.024344320873288041966754378377, −7.63438799443331079810058605739, −5.76092044500818301723307364654, −3.48199102173636645566001738889, −1.72135155805269593212373944218, −0.30155456506777471712669598924, 2.07864268281419510112254773437, 5.30948659649610856646916557787, 6.02674966728915789147661263564, 7.952970462032991196312843567774, 8.712082135035681707997389384282, 9.834982419420634104762891978125, 11.58926066298391566516557329544, 12.76790349526225549402779919841, 14.66809613383004309986494917670, 15.14091401766533949462407708774

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