Properties

Label 2-43-43.18-c6-0-10
Degree $2$
Conductor $43$
Sign $0.663 + 0.748i$
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  + (−2.84 + 5.90i)2-s + (−20.1 + 29.6i)3-s + (13.1 + 16.4i)4-s + (−116. − 125. i)5-s + (−117. − 203. i)6-s + (−46.7 − 27.0i)7-s + (−543. + 124. i)8-s + (−202. − 516. i)9-s + (1.07e3 − 330. i)10-s + (−81.3 + 102. i)11-s + (−752. + 56.4i)12-s + (3.50e3 + 1.08e3i)13-s + (292. − 199. i)14-s + (6.05e3 − 913. i)15-s + (512. − 2.24e3i)16-s + (−7.06e3 − 6.55e3i)17-s + ⋯
L(s)  = 1  + (−0.355 + 0.737i)2-s + (−0.747 + 1.09i)3-s + (0.205 + 0.257i)4-s + (−0.930 − 1.00i)5-s + (−0.543 − 0.941i)6-s + (−0.136 − 0.0787i)7-s + (−1.06 + 0.242i)8-s + (−0.278 − 0.708i)9-s + (1.07 − 0.330i)10-s + (−0.0611 + 0.0766i)11-s + (−0.435 + 0.0326i)12-s + (1.59 + 0.492i)13-s + (0.106 − 0.0726i)14-s + (1.79 − 0.270i)15-s + (0.125 − 0.548i)16-s + (−1.43 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.663 + 0.748i$
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: \((2,\ 43,\ (\ :3),\ 0.663 + 0.748i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.200709 - 0.0902502i\)
\(L(\frac12)\) \(\approx\) \(0.200709 - 0.0902502i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (4.25e4 + 6.71e4i)T \)
good2 \( 1 + (2.84 - 5.90i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (20.1 - 29.6i)T + (-266. - 678. i)T^{2} \)
5 \( 1 + (116. + 125. i)T + (-1.16e3 + 1.55e4i)T^{2} \)
7 \( 1 + (46.7 + 27.0i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (81.3 - 102. i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (-3.50e3 - 1.08e3i)T + (3.98e6 + 2.71e6i)T^{2} \)
17 \( 1 + (7.06e3 + 6.55e3i)T + (1.80e6 + 2.40e7i)T^{2} \)
19 \( 1 + (-1.78e3 - 700. i)T + (3.44e7 + 3.19e7i)T^{2} \)
23 \( 1 + (-1.03e4 - 1.56e3i)T + (1.41e8 + 4.36e7i)T^{2} \)
29 \( 1 + (-4.42e3 - 6.49e3i)T + (-2.17e8 + 5.53e8i)T^{2} \)
31 \( 1 + (3.56e3 + 4.75e4i)T + (-8.77e8 + 1.32e8i)T^{2} \)
37 \( 1 + (5.28e4 - 3.04e4i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + (5.10e4 + 2.45e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (-1.11e4 - 1.39e4i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (8.35e4 - 2.57e4i)T + (1.83e10 - 1.24e10i)T^{2} \)
59 \( 1 + (-7.63e4 + 3.34e5i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (-4.37e3 - 327. i)T + (5.09e10 + 7.67e9i)T^{2} \)
67 \( 1 + (4.76e4 - 1.21e5i)T + (-6.63e10 - 6.15e10i)T^{2} \)
71 \( 1 + (-3.39e4 - 2.25e5i)T + (-1.22e11 + 3.77e10i)T^{2} \)
73 \( 1 + (1.02e5 - 3.31e5i)T + (-1.25e11 - 8.52e10i)T^{2} \)
79 \( 1 + (-1.16e4 + 2.01e4i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-1.14e5 - 7.80e4i)T + (1.19e11 + 3.04e11i)T^{2} \)
89 \( 1 + (2.06e5 - 3.02e5i)T + (-1.81e11 - 4.62e11i)T^{2} \)
97 \( 1 + (5.97e5 - 7.48e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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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

−15.44711538881163712133014588960, −13.32565516363993149361869055073, −11.75517004936428958826001535485, −11.16593357857420852441570152449, −9.285132998328211933275439725151, −8.380824550179159889302230518771, −6.77761812512352598904142952548, −5.14538327861454803486031781877, −3.85096880745339757440953677910, −0.13352694872564287356239841525, 1.42413350484246717985873888468, 3.30360286882312405044690678091, 6.14670820831589169354788651756, 6.91171569172452754590024625236, 8.572372944708835836865576010367, 10.76504281134940744259267720363, 11.05624475741305815461683013606, 12.18436463229160168406607134775, 13.26637343776891427942158084445, 15.02681380103509181277577843917

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