Properties

Degree 2
Conductor 43
Sign $-0.885 - 0.464i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.22 + 2.78i)2-s + (−37.3 − 5.62i)3-s + (25.6 − 112. i)4-s + (39.7 + 27.1i)5-s + (−67.1 − 116. i)6-s + (−76.5 + 132. i)7-s + (780. − 376. i)8-s + (−730. − 225. i)9-s + (12.8 + 171. i)10-s + (247. + 1.08e3i)11-s + (−1.58e3 + 4.04e3i)12-s + (−542. + 7.23e3i)13-s + (−539. + 81.3i)14-s + (−1.33e3 − 1.23e3i)15-s + (−1.05e4 − 5.06e3i)16-s + (−2.38e4 + 1.62e4i)17-s + ⋯
L(s)  = 1  + (0.196 + 0.246i)2-s + (−0.797 − 0.120i)3-s + (0.200 − 0.878i)4-s + (0.142 + 0.0970i)5-s + (−0.126 − 0.219i)6-s + (−0.0843 + 0.146i)7-s + (0.539 − 0.259i)8-s + (−0.333 − 0.102i)9-s + (0.00405 + 0.0541i)10-s + (0.0560 + 0.245i)11-s + (−0.265 + 0.676i)12-s + (−0.0684 + 0.913i)13-s + (−0.0525 + 0.00792i)14-s + (−0.101 − 0.0945i)15-s + (−0.641 − 0.309i)16-s + (−1.17 + 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.885 - 0.464i$
motivic weight  =  \(7\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ -0.885 - 0.464i)\)
\(L(4)\)  \(\approx\)  \(0.0504570 + 0.204914i\)
\(L(\frac12)\)  \(\approx\)  \(0.0504570 + 0.204914i\)
\(L(\frac{9}{2})\)   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 \( 1 + (-2.09e5 + 4.77e5i)T \)
good2 \( 1 + (-2.22 - 2.78i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (37.3 + 5.62i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (-39.7 - 27.1i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (76.5 - 132. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-247. - 1.08e3i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (542. - 7.23e3i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (2.38e4 - 1.62e4i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (3.33e4 - 1.02e4i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (8.25e3 - 7.65e3i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (1.52e4 - 2.29e3i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (1.02e5 - 2.60e5i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (-1.00e5 - 1.74e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (-1.40e5 - 1.76e5i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (-5.53e4 + 2.42e5i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (1.15e5 + 1.53e6i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (1.58e6 + 7.64e5i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (5.73e5 + 1.46e6i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (2.80e6 - 8.64e5i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (1.85e6 + 1.72e6i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (6.58e4 - 8.78e5i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (-3.30e5 + 5.72e5i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (4.32e6 + 6.51e5i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (2.92e6 + 4.40e5i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (1.26e6 + 5.55e6i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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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

−14.90791388574496387147943518882, −13.97381315671120507892708237332, −12.51267740594037504939032858067, −11.26101419951153423952600721861, −10.36335943791961355266620050800, −8.852360920081351997899451717813, −6.72733255636683438511812307845, −6.00981824957598350160155523972, −4.54730118722113270138177411406, −1.86643838328272151411547539774, 0.085151682064114447641170651242, 2.60132222942182328063207666447, 4.37922781584953325772048809008, 5.93821704561760724505707341787, 7.49664451758935891113797056677, 8.931406074280059904575431069909, 10.78519898391100389159259721997, 11.46003290069887233598502511501, 12.71876578837547864233963911494, 13.58589650621979105289626629813

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