Properties

Label 2-43-43.18-c8-0-15
Degree $2$
Conductor $43$
Sign $0.987 - 0.156i$
Analytic cond. $17.5172$
Root an. cond. $4.18536$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.645 − 1.34i)2-s + (42.1 − 61.8i)3-s + (158. + 198. i)4-s + (−171. − 184. i)5-s + (−55.6 − 96.4i)6-s + (2.10e3 + 1.21e3i)7-s + (739. − 168. i)8-s + (350. + 891. i)9-s + (−357. + 110. i)10-s + (−7.36e3 + 9.23e3i)11-s + (1.89e4 − 1.41e3i)12-s + (2.84e4 + 8.77e3i)13-s + (2.98e3 − 2.03e3i)14-s + (−1.86e4 + 2.80e3i)15-s + (−1.42e4 + 6.22e4i)16-s + (−1.14e5 − 1.05e5i)17-s + ⋯
L(s)  = 1  + (0.0403 − 0.0837i)2-s + (0.520 − 0.763i)3-s + (0.618 + 0.775i)4-s + (−0.273 − 0.295i)5-s + (−0.0429 − 0.0744i)6-s + (0.874 + 0.505i)7-s + (0.180 − 0.0412i)8-s + (0.0533 + 0.135i)9-s + (−0.0357 + 0.0110i)10-s + (−0.502 + 0.630i)11-s + (0.913 − 0.0684i)12-s + (0.995 + 0.307i)13-s + (0.0776 − 0.0529i)14-s + (−0.367 + 0.0554i)15-s + (−0.216 + 0.949i)16-s + (−1.36 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.987 - 0.156i$
Analytic conductor: \(17.5172\)
Root analytic conductor: \(4.18536\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :4),\ 0.987 - 0.156i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.80578 + 0.220550i\)
\(L(\frac12)\) \(\approx\) \(2.80578 + 0.220550i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-3.41e6 - 1.16e4i)T \)
good2 \( 1 + (-0.645 + 1.34i)T + (-159. - 200. i)T^{2} \)
3 \( 1 + (-42.1 + 61.8i)T + (-2.39e3 - 6.10e3i)T^{2} \)
5 \( 1 + (171. + 184. i)T + (-2.91e4 + 3.89e5i)T^{2} \)
7 \( 1 + (-2.10e3 - 1.21e3i)T + (2.88e6 + 4.99e6i)T^{2} \)
11 \( 1 + (7.36e3 - 9.23e3i)T + (-4.76e7 - 2.08e8i)T^{2} \)
13 \( 1 + (-2.84e4 - 8.77e3i)T + (6.73e8 + 4.59e8i)T^{2} \)
17 \( 1 + (1.14e5 + 1.05e5i)T + (5.21e8 + 6.95e9i)T^{2} \)
19 \( 1 + (-1.79e5 - 7.03e4i)T + (1.24e10 + 1.15e10i)T^{2} \)
23 \( 1 + (-3.32e5 - 5.01e4i)T + (7.48e10 + 2.30e10i)T^{2} \)
29 \( 1 + (-2.29e5 - 3.36e5i)T + (-1.82e11 + 4.65e11i)T^{2} \)
31 \( 1 + (-2.32e4 - 3.09e5i)T + (-8.43e11 + 1.27e11i)T^{2} \)
37 \( 1 + (-1.03e6 + 5.98e5i)T + (1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + (1.22e6 + 5.91e5i)T + (4.97e12 + 6.24e12i)T^{2} \)
47 \( 1 + (5.47e6 + 6.87e6i)T + (-5.29e12 + 2.32e13i)T^{2} \)
53 \( 1 + (1.38e7 - 4.27e6i)T + (5.14e13 - 3.50e13i)T^{2} \)
59 \( 1 + (2.03e6 - 8.91e6i)T + (-1.32e14 - 6.37e13i)T^{2} \)
61 \( 1 + (1.87e7 + 1.40e6i)T + (1.89e14 + 2.85e13i)T^{2} \)
67 \( 1 + (1.09e6 - 2.78e6i)T + (-2.97e14 - 2.76e14i)T^{2} \)
71 \( 1 + (-6.81e6 - 4.51e7i)T + (-6.17e14 + 1.90e14i)T^{2} \)
73 \( 1 + (-1.33e7 + 4.34e7i)T + (-6.66e14 - 4.54e14i)T^{2} \)
79 \( 1 + (7.94e6 - 1.37e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + (9.53e6 + 6.49e6i)T + (8.22e14 + 2.09e15i)T^{2} \)
89 \( 1 + (1.38e7 - 2.02e7i)T + (-1.43e15 - 3.66e15i)T^{2} \)
97 \( 1 + (-9.98e7 + 1.25e8i)T + (-1.74e15 - 7.64e15i)T^{2} \)
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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

−13.98685422135374097746315355405, −13.01958589249673137649784220980, −11.94867133693540861009695892212, −11.02403988065937448978071313508, −8.852151186343395676511127441500, −7.88436520580501021197518686291, −6.95047630117469797480126861234, −4.78781511629505870653152825995, −2.79824840465911984875808685096, −1.58063103327394437434713825413, 1.16708956722798179031389611005, 3.16040280582910824219119286122, 4.73659381394076777521920337624, 6.37746213511590946608527145730, 7.936599281453029367509920937303, 9.336428658420790900920972797121, 10.88261521328692355444472341884, 11.10537191327118485895577516291, 13.33809223318048311398961151695, 14.48131897874547333945857977201

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