Properties

Degree 2
Conductor 43
Sign $-0.773 - 0.633i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.10 − 12.6i)2-s + (64.4 − 133. i)3-s + (36.2 − 45.4i)4-s + (363. + 82.8i)5-s − 2.08e3·6-s + 1.93e3i·7-s + (−4.30e3 − 983. i)8-s + (−9.64e3 − 1.20e4i)9-s + (−1.16e3 − 5.10e3i)10-s + (−6.27e3 − 7.87e3i)11-s + (−3.74e3 − 7.77e3i)12-s + (−8.78e3 + 3.85e4i)13-s + (2.44e4 − 1.17e4i)14-s + (3.44e4 − 4.32e4i)15-s + (1.05e4 + 4.61e4i)16-s + (−3.16e4 − 1.38e5i)17-s + ⋯
L(s)  = 1  + (−0.381 − 0.792i)2-s + (0.795 − 1.65i)3-s + (0.141 − 0.177i)4-s + (0.580 + 0.132i)5-s − 1.61·6-s + 0.804i·7-s + (−1.05 − 0.240i)8-s + (−1.47 − 1.84i)9-s + (−0.116 − 0.510i)10-s + (−0.428 − 0.537i)11-s + (−0.180 − 0.374i)12-s + (−0.307 + 1.34i)13-s + (0.637 − 0.306i)14-s + (0.680 − 0.853i)15-s + (0.160 + 0.703i)16-s + (−0.379 − 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.773 - 0.633i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.773 - 0.633i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.573151 + 1.60558i\)
\(L(\frac12)\)  \(\approx\)  \(0.573151 + 1.60558i\)
\(L(5)\)   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 \( 1 + (-3.37e6 - 5.37e5i)T \)
good2 \( 1 + (6.10 + 12.6i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (-64.4 + 133. i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (-363. - 82.8i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 - 1.93e3iT - 5.76e6T^{2} \)
11 \( 1 + (6.27e3 + 7.87e3i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (8.78e3 - 3.85e4i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (3.16e4 + 1.38e5i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (5.56e4 + 4.43e4i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (-5.77e4 - 7.24e4i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (-2.05e4 - 4.26e4i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (-1.41e6 + 6.82e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 + 2.69e5iT - 3.51e12T^{2} \)
41 \( 1 + (-1.04e6 + 5.03e5i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (-6.04e6 + 7.57e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (1.95e5 + 8.58e5i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (7.90e5 + 3.46e6i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (-4.57e5 + 9.50e5i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (-1.63e7 + 2.05e7i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (-1.80e7 - 1.44e7i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (3.86e7 + 8.82e6i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 + 5.81e7T + 1.51e15T^{2} \)
83 \( 1 + (1.06e7 + 5.14e6i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (1.03e7 - 2.15e7i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (-2.14e7 - 2.68e7i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−13.48123179676743298509858581772, −12.14028624359078209726477262628, −11.46595949383953001645839656105, −9.521835032091012772125851145708, −8.714802142308836699589558634053, −7.05459511175965948416542587056, −5.98479972365223653916033130383, −2.61427420695507407223603437936, −2.15324722455409050575357419075, −0.61748336854332717451651136151, 2.74613766767171130918583224071, 4.21014377217289947096923617595, 5.78930603494362611926866944141, 7.75954213164394587944843395876, 8.673837019008508168981250955913, 10.00800527184839896872356305406, 10.64825222394712631449791073072, 12.83285252163462509868029333142, 14.27692089785149684795076831394, 15.25927825101689037264153574831

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