Properties

Degree 2
Conductor 43
Sign $0.883 + 0.468i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.178i·2-s + (33.4 − 19.3i)3-s + 63.9·4-s + (21.1 − 12.1i)5-s + (−3.45 − 5.97i)6-s + (409. + 236. i)7-s − 22.8i·8-s + (380. − 659. i)9-s + (−2.17 − 3.77i)10-s − 2.21e3·11-s + (2.13e3 − 1.23e3i)12-s + (−1.29e3 + 2.24e3i)13-s + (42.2 − 73.1i)14-s + (470. − 814. i)15-s + 4.08e3·16-s + (4.48e3 − 7.76e3i)17-s + ⋯
L(s)  = 1  − 0.0223i·2-s + (1.23 − 0.714i)3-s + 0.999·4-s + (0.168 − 0.0974i)5-s + (−0.0159 − 0.0276i)6-s + (1.19 + 0.688i)7-s − 0.0447i·8-s + (0.522 − 0.904i)9-s + (−0.00217 − 0.00377i)10-s − 1.66·11-s + (1.23 − 0.714i)12-s + (−0.590 + 1.02i)13-s + (0.0153 − 0.0266i)14-s + (0.139 − 0.241i)15-s + 0.998·16-s + (0.912 − 1.58i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.883 + 0.468i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.883 + 0.468i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(3.12515 - 0.777646i\)
\(L(\frac12)\)  \(\approx\)  \(3.12515 - 0.777646i\)
\(L(4)\)   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 \( 1 + (4.31e4 - 6.67e4i)T \)
good2 \( 1 + 0.178iT - 64T^{2} \)
3 \( 1 + (-33.4 + 19.3i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (-21.1 + 12.1i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-409. - 236. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + 2.21e3T + 1.77e6T^{2} \)
13 \( 1 + (1.29e3 - 2.24e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (-4.48e3 + 7.76e3i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (699. - 403. i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (2.74e3 + 4.74e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (1.47e4 + 8.52e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (2.25e4 + 3.90e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (8.88e3 - 5.12e3i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 1.69e4T + 4.75e9T^{2} \)
47 \( 1 - 7.30e4T + 1.07e10T^{2} \)
53 \( 1 + (-9.62e4 - 1.66e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + 1.37e5T + 4.21e10T^{2} \)
61 \( 1 + (3.54e5 + 2.04e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-1.59e5 - 2.75e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (-4.49e5 - 2.59e5i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (-1.89e5 - 1.09e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-3.84e5 + 6.66e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-1.26e5 - 2.18e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (4.34e5 - 2.51e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 2.80e5T + 8.32e11T^{2} \)
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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

−14.60430045395242671850211306215, −13.59081112095010774188232223720, −12.25794048684466737669665770913, −11.21268310598385895752557853997, −9.475060356794030300901803199353, −7.947848976466934167765325026974, −7.41694769761774211374716682064, −5.35418517349218127328495169393, −2.67876970017467860119020730749, −1.89713954036931418794631167520, 2.03265488298720026393638417384, 3.43541113094124009796886098892, 5.33257664097387648681427007021, 7.67247385357316396059843256589, 8.172701466475694140904071684820, 10.31092129749443475297924738501, 10.61067089388471424241547919877, 12.51901982247667929777265336324, 13.98487458790404620409537615809, 14.95824721873843101440194035004

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