Properties

Degree 2
Conductor 43
Sign $-0.987 + 0.157i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.21 + 6.66i)2-s + (13.4 + 1.00i)3-s + (−24.1 − 30.2i)4-s + (−6.13 + 19.8i)5-s + (−49.7 + 86.1i)6-s + (−69.5 + 40.1i)7-s + (164. − 37.4i)8-s + (98.7 + 14.8i)9-s + (−112. − 104. i)10-s + (−66.3 + 83.1i)11-s + (−293. − 430. i)12-s + (54.4 − 50.4i)13-s + (−44.3 − 592. i)14-s + (−102. + 260. i)15-s + (−139. + 609. i)16-s + (499. − 153. i)17-s + ⋯
L(s)  = 1  + (−0.802 + 1.66i)2-s + (1.48 + 0.111i)3-s + (−1.50 − 1.89i)4-s + (−0.245 + 0.795i)5-s + (−1.38 + 2.39i)6-s + (−1.41 + 0.818i)7-s + (2.56 − 0.585i)8-s + (1.21 + 0.183i)9-s + (−1.12 − 1.04i)10-s + (−0.547 + 0.687i)11-s + (−2.03 − 2.98i)12-s + (0.321 − 0.298i)13-s + (−0.226 − 3.02i)14-s + (−0.454 + 1.15i)15-s + (−0.543 + 2.38i)16-s + (1.72 − 0.532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.987 + 0.157i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ -0.987 + 0.157i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.0877227 - 1.10471i\)
\(L(\frac12)\)  \(\approx\)  \(0.0877227 - 1.10471i\)
\(L(3)\)   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 \( 1 + (-762. - 1.68e3i)T \)
good2 \( 1 + (3.21 - 6.66i)T + (-9.97 - 12.5i)T^{2} \)
3 \( 1 + (-13.4 - 1.00i)T + (80.0 + 12.0i)T^{2} \)
5 \( 1 + (6.13 - 19.8i)T + (-516. - 352. i)T^{2} \)
7 \( 1 + (69.5 - 40.1i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (66.3 - 83.1i)T + (-3.25e3 - 1.42e4i)T^{2} \)
13 \( 1 + (-54.4 + 50.4i)T + (2.13e3 - 2.84e4i)T^{2} \)
17 \( 1 + (-499. + 153. i)T + (6.90e4 - 4.70e4i)T^{2} \)
19 \( 1 + (-54.5 - 362. i)T + (-1.24e5 + 3.84e4i)T^{2} \)
23 \( 1 + (-51.1 - 130. i)T + (-2.05e5 + 1.90e5i)T^{2} \)
29 \( 1 + (514. - 38.5i)T + (6.99e5 - 1.05e5i)T^{2} \)
31 \( 1 + (-495. + 337. i)T + (3.37e5 - 8.59e5i)T^{2} \)
37 \( 1 + (350. + 202. i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + (-2.38e3 - 1.14e3i)T + (1.76e6 + 2.20e6i)T^{2} \)
47 \( 1 + (998. + 1.25e3i)T + (-1.08e6 + 4.75e6i)T^{2} \)
53 \( 1 + (801. + 743. i)T + (5.89e5 + 7.86e6i)T^{2} \)
59 \( 1 + (-709. + 3.10e3i)T + (-1.09e7 - 5.25e6i)T^{2} \)
61 \( 1 + (1.04e3 - 1.52e3i)T + (-5.05e6 - 1.28e7i)T^{2} \)
67 \( 1 + (2.28e3 - 344. i)T + (1.92e7 - 5.93e6i)T^{2} \)
71 \( 1 + (-7.45e3 - 2.92e3i)T + (1.86e7 + 1.72e7i)T^{2} \)
73 \( 1 + (1.30e3 + 1.40e3i)T + (-2.12e6 + 2.83e7i)T^{2} \)
79 \( 1 + (1.28e3 + 2.22e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (62.5 - 835. i)T + (-4.69e7 - 7.07e6i)T^{2} \)
89 \( 1 + (4.91e3 + 368. i)T + (6.20e7 + 9.35e6i)T^{2} \)
97 \( 1 + (-6.21e3 + 7.79e3i)T + (-1.96e7 - 8.63e7i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.65931673956395244222246485203, −14.91141865910550968471408235986, −14.21059541322005751548708881883, −12.85922718899713421049560101332, −9.956766741444998446785988393799, −9.472198053802078917660042085446, −8.110651049606776944256221385787, −7.28129591430224291882560476625, −5.83546593068755692352555295933, −3.17252082603948777070166523878, 0.829900635375572508904126288492, 2.91675423688062201522285250134, 3.82717084614202676810273969829, 7.70573053964371948543007501912, 8.767889486875500169980294766231, 9.572271327105246792751453472813, 10.63656335869557881899933783357, 12.41151682552864558507165508694, 13.13702097354082753554930583859, 13.94410681844626974878063123886

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