Properties

Label 2-43-43.18-c6-0-17
Degree $2$
Conductor $43$
Sign $-0.815 + 0.578i$
Analytic cond. $9.89232$
Root an. cond. $3.14520$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.71 − 5.63i)2-s + (26.7 − 39.2i)3-s + (15.4 + 19.4i)4-s + (−88.5 − 95.4i)5-s + (−148. − 257. i)6-s + (−56.9 − 32.8i)7-s + (542. − 123. i)8-s + (−557. − 1.41e3i)9-s + (−778. + 240. i)10-s + (−698. + 876. i)11-s + (1.17e3 − 88.1i)12-s + (2.66e3 + 821. i)13-s + (−340. + 231. i)14-s + (−6.11e3 + 921. i)15-s + (420. − 1.84e3i)16-s + (573. + 532. i)17-s + ⋯
L(s)  = 1  + (0.339 − 0.704i)2-s + (0.990 − 1.45i)3-s + (0.241 + 0.303i)4-s + (−0.708 − 0.763i)5-s + (−0.687 − 1.19i)6-s + (−0.166 − 0.0958i)7-s + (1.05 − 0.241i)8-s + (−0.764 − 1.94i)9-s + (−0.778 + 0.240i)10-s + (−0.525 + 0.658i)11-s + (0.680 − 0.0509i)12-s + (1.21 + 0.374i)13-s + (−0.123 + 0.0845i)14-s + (−1.81 + 0.273i)15-s + (0.102 − 0.449i)16-s + (0.116 + 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.815 + 0.578i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ -0.815 + 0.578i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.821345 - 2.57754i\)
\(L(\frac12)\) \(\approx\) \(0.821345 - 2.57754i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-7.13e4 + 3.50e4i)T \)
good2 \( 1 + (-2.71 + 5.63i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (-26.7 + 39.2i)T + (-266. - 678. i)T^{2} \)
5 \( 1 + (88.5 + 95.4i)T + (-1.16e3 + 1.55e4i)T^{2} \)
7 \( 1 + (56.9 + 32.8i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (698. - 876. i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (-2.66e3 - 821. i)T + (3.98e6 + 2.71e6i)T^{2} \)
17 \( 1 + (-573. - 532. i)T + (1.80e6 + 2.40e7i)T^{2} \)
19 \( 1 + (8.77e3 + 3.44e3i)T + (3.44e7 + 3.19e7i)T^{2} \)
23 \( 1 + (-1.36e4 - 2.04e3i)T + (1.41e8 + 4.36e7i)T^{2} \)
29 \( 1 + (-2.79e3 - 4.09e3i)T + (-2.17e8 + 5.53e8i)T^{2} \)
31 \( 1 + (-149. - 1.99e3i)T + (-8.77e8 + 1.32e8i)T^{2} \)
37 \( 1 + (-6.75e3 + 3.90e3i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + (-8.34e4 - 4.01e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (-4.76e4 - 5.97e4i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (-2.22e5 + 6.87e4i)T + (1.83e10 - 1.24e10i)T^{2} \)
59 \( 1 + (-1.83e4 + 8.02e4i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (4.27e5 + 3.20e4i)T + (5.09e10 + 7.67e9i)T^{2} \)
67 \( 1 + (1.80e5 - 4.60e5i)T + (-6.63e10 - 6.15e10i)T^{2} \)
71 \( 1 + (4.66e4 + 3.09e5i)T + (-1.22e11 + 3.77e10i)T^{2} \)
73 \( 1 + (-2.25e3 + 7.29e3i)T + (-1.25e11 - 8.52e10i)T^{2} \)
79 \( 1 + (4.56e5 - 7.91e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-8.89e5 - 6.06e5i)T + (1.19e11 + 3.04e11i)T^{2} \)
89 \( 1 + (6.83e5 - 1.00e6i)T + (-1.81e11 - 4.62e11i)T^{2} \)
97 \( 1 + (4.69e5 - 5.88e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.64455749666590654613788750905, −12.87140289299297849640503604782, −12.33968599693932564417955634252, −11.04085382092269750776177154815, −8.835332568218674241764126784434, −7.901678919451706409982554918869, −6.81325818417966656595322195228, −4.07686269505825046654443027985, −2.55522005206373382095264117671, −1.14295879929712903980452014858, 2.96988571384636050912826975653, 4.25088940777909049612745426548, 5.90465169350027521812282465744, 7.68591053163605387112414507734, 8.840177560651591188271852076389, 10.59982074103251681275107993306, 10.89563590672698636316240977137, 13.43661907360024800474695018930, 14.51425213195448467825008733424, 15.23268404357778647087201828236

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