Properties

Degree 2
Conductor 43
Sign $-0.892 + 0.450i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.38i·2-s + (−8.75 − 5.05i)3-s + 9.44·4-s + (10.2 + 5.93i)5-s + (−37.3 + 64.6i)6-s + (367. − 211. i)7-s − 542. i·8-s + (−313. − 542. i)9-s + (43.8 − 75.9i)10-s − 1.64e3·11-s + (−82.6 − 47.7i)12-s + (1.10e3 + 1.90e3i)13-s + (−1.56e3 − 2.71e3i)14-s + (−60.0 − 104. i)15-s − 3.40e3·16-s + (−2.12e3 − 3.68e3i)17-s + ⋯
L(s)  = 1  − 0.923i·2-s + (−0.324 − 0.187i)3-s + 0.147·4-s + (0.0822 + 0.0475i)5-s + (−0.172 + 0.299i)6-s + (1.07 − 0.617i)7-s − 1.05i·8-s + (−0.429 − 0.744i)9-s + (0.0438 − 0.0759i)10-s − 1.23·11-s + (−0.0478 − 0.0276i)12-s + (0.501 + 0.868i)13-s + (−0.570 − 0.988i)14-s + (−0.0177 − 0.0308i)15-s − 0.830·16-s + (−0.433 − 0.750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.892 + 0.450i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.892 + 0.450i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.364933 - 1.53370i\)
\(L(\frac12)\)  \(\approx\)  \(0.364933 - 1.53370i\)
\(L(4)\)   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 \( 1 + (-4.17e4 - 6.76e4i)T \)
good2 \( 1 + 7.38iT - 64T^{2} \)
3 \( 1 + (8.75 + 5.05i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (-10.2 - 5.93i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (-367. + 211. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + 1.64e3T + 1.77e6T^{2} \)
13 \( 1 + (-1.10e3 - 1.90e3i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (2.12e3 + 3.68e3i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (2.14e3 + 1.23e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-1.16e3 + 2.01e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (1.27e4 - 7.38e3i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-2.70e4 + 4.68e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-5.82e4 - 3.36e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 1.24e5T + 4.75e9T^{2} \)
47 \( 1 + 8.27e4T + 1.07e10T^{2} \)
53 \( 1 + (6.94e4 - 1.20e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 - 2.18e5T + 4.21e10T^{2} \)
61 \( 1 + (5.65e4 - 3.26e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.83e5 + 3.17e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (-1.19e5 + 6.92e4i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (-4.14e5 + 2.39e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-4.53e5 - 7.84e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (1.55e5 - 2.69e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (5.79e5 + 3.34e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 7.31e5T + 8.32e11T^{2} \)
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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

−13.96247935753790803751441883154, −12.78925489139897753043845727424, −11.42829178259980636769722549315, −11.05199920598479943548428787833, −9.581787166839238018838872815442, −7.81666234747106449484629666922, −6.32651203942092908729807474965, −4.35865674812172956842829807557, −2.44417620275127466104455088996, −0.76102889027898227518027293444, 2.27270500235600648861689045531, 5.10482042492058094078080832520, 5.84500840878872985591671513301, 7.77412793330304934466343178723, 8.424737121177421498167118431778, 10.65635773142536868124404653894, 11.34886821747356614681277622899, 13.01032623120649951009925089197, 14.39182271740937164718641214298, 15.39078595049837747532346780930

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