Properties

Degree 2
Conductor 43
Sign $-0.533 + 0.845i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.6i·2-s + (−24.0 + 13.8i)3-s − 48.8·4-s + (200. − 115. i)5-s + (147. + 255. i)6-s + (396. + 228. i)7-s − 161. i·8-s + (20.9 − 36.2i)9-s + (−1.23e3 − 2.13e3i)10-s − 1.24e3·11-s + (1.17e3 − 677. i)12-s + (1.54e3 − 2.68e3i)13-s + (2.43e3 − 4.21e3i)14-s + (−3.21e3 + 5.57e3i)15-s − 4.83e3·16-s + (3.59e3 − 6.22e3i)17-s + ⋯
L(s)  = 1  − 1.32i·2-s + (−0.890 + 0.514i)3-s − 0.762·4-s + (1.60 − 0.926i)5-s + (0.682 + 1.18i)6-s + (1.15 + 0.667i)7-s − 0.315i·8-s + (0.0287 − 0.0497i)9-s + (−1.23 − 2.13i)10-s − 0.934·11-s + (0.679 − 0.392i)12-s + (0.705 − 1.22i)13-s + (0.885 − 1.53i)14-s + (−0.952 + 1.65i)15-s − 1.18·16-s + (0.731 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.533 + 0.845i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.533 + 0.845i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.859130 - 1.55821i\)
\(L(\frac12)\)  \(\approx\)  \(0.859130 - 1.55821i\)
\(L(4)\)   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 \( 1 + (6.33e4 + 4.80e4i)T \)
good2 \( 1 + 10.6iT - 64T^{2} \)
3 \( 1 + (24.0 - 13.8i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (-200. + 115. i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-396. - 228. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + 1.24e3T + 1.77e6T^{2} \)
13 \( 1 + (-1.54e3 + 2.68e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (-3.59e3 + 6.22e3i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (4.53e3 - 2.61e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-4.97e3 - 8.61e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-2.16e4 - 1.24e4i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (1.08e4 + 1.87e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (9.19e3 - 5.30e3i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 4.43e4T + 4.75e9T^{2} \)
47 \( 1 + 2.60e3T + 1.07e10T^{2} \)
53 \( 1 + (-1.27e5 - 2.21e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 - 1.67e5T + 4.21e10T^{2} \)
61 \( 1 + (-1.27e5 - 7.38e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-8.89e4 - 1.54e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (-2.48e5 - 1.43e5i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (5.92e4 + 3.41e4i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (1.60e5 - 2.77e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-1.88e5 - 3.25e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (9.23e5 - 5.33e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 5.08e5T + 8.32e11T^{2} \)
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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

−13.79054066863617214879202648302, −12.84593658297787288949630350050, −11.75353240970042175532713684224, −10.64270015559851374271901923645, −9.935121740319557674851616815410, −8.523717341611901136037962655188, −5.57368140998831370069635352716, −5.05153562368123202337396244489, −2.43090363083072000310601479782, −1.02356687059889190350785519298, 1.78719037394740633001285183268, 5.15613597148927192908616465078, 6.26233613515614854869621537047, 6.90160293329384149683652554691, 8.459943158788673128195636707494, 10.42850075187848669601389705126, 11.27088671074671472763631173514, 13.21298043192071124473407665113, 14.22125258114499459774604454260, 14.84387470082635279870742304589

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