Properties

Degree 2
Conductor 43
Sign $0.678 + 0.734i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12.8i·2-s + (−27.1 + 15.6i)3-s − 100.·4-s + (−154. + 89.2i)5-s + (−201. − 348. i)6-s + (237. + 137. i)7-s − 465. i·8-s + (128. − 222. i)9-s + (−1.14e3 − 1.98e3i)10-s + 1.24e3·11-s + (2.72e3 − 1.57e3i)12-s + (−1.13e3 + 1.96e3i)13-s + (−1.75e3 + 3.04e3i)14-s + (2.80e3 − 4.85e3i)15-s − 449.·16-s + (−125. + 218. i)17-s + ⋯
L(s)  = 1  + 1.60i·2-s + (−1.00 + 0.581i)3-s − 1.56·4-s + (−1.23 + 0.713i)5-s + (−0.931 − 1.61i)6-s + (0.692 + 0.400i)7-s − 0.909i·8-s + (0.175 − 0.304i)9-s + (−1.14 − 1.98i)10-s + 0.938·11-s + (1.57 − 0.911i)12-s + (−0.516 + 0.894i)13-s + (−0.641 + 1.11i)14-s + (0.829 − 1.43i)15-s − 0.109·16-s + (−0.0256 + 0.0444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.678 + 0.734i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.678 + 0.734i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.453448 - 0.198388i\)
\(L(\frac12)\)  \(\approx\)  \(0.453448 - 0.198388i\)
\(L(4)\)   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 \( 1 + (6.28e4 - 4.87e4i)T \)
good2 \( 1 - 12.8iT - 64T^{2} \)
3 \( 1 + (27.1 - 15.6i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (154. - 89.2i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-237. - 137. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 - 1.24e3T + 1.77e6T^{2} \)
13 \( 1 + (1.13e3 - 1.96e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (125. - 218. i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-5.83e3 + 3.37e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-6.60e3 - 1.14e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (2.13e4 + 1.23e4i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (9.21e3 + 1.59e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (6.18e4 - 3.56e4i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 1.01e5T + 4.75e9T^{2} \)
47 \( 1 - 826.T + 1.07e10T^{2} \)
53 \( 1 + (-1.23e5 - 2.13e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 - 2.87e5T + 4.21e10T^{2} \)
61 \( 1 + (3.73e5 + 2.15e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (6.87e4 + 1.19e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (-9.04e4 - 5.22e4i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (-2.86e5 - 1.65e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (4.19e5 - 7.26e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (8.97e4 + 1.55e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (4.50e5 - 2.60e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.59e6T + 8.32e11T^{2} \)
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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

−15.63556551617557884828268510855, −15.02267869762349836141719457589, −14.03205420811605503979030011357, −11.69361083718740659709540233441, −11.30531095206691585152064499693, −9.283705362531282553244111393000, −7.77732600122566017850370795086, −6.79757601262618259833219131294, −5.37820407995069153785643380205, −4.21479254981673830814792804715, 0.31307186102944976767628802142, 1.24781001965846931696284210368, 3.70197470048134905417967414003, 5.05608910785665141474097940591, 7.32550256056503647566033473890, 8.874669249576397264264272183869, 10.60320850291246965499644261846, 11.55644052622156604509319804601, 12.14204213213229448506640999233, 12.86783252839938696385725290195

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