Properties

Label 2-43-43.7-c6-0-9
Degree $2$
Conductor $43$
Sign $-0.0958 + 0.995i$
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  − 4.28i·2-s + (−41.9 + 24.2i)3-s + 45.6·4-s + (−171. + 98.7i)5-s + (103. + 179. i)6-s + (223. + 129. i)7-s − 469. i·8-s + (807. − 1.39e3i)9-s + (423. + 733. i)10-s − 1.62e3·11-s + (−1.91e3 + 1.10e3i)12-s + (1.10e3 − 1.90e3i)13-s + (553. − 959. i)14-s + (4.78e3 − 8.27e3i)15-s + 905.·16-s + (−209. + 362. i)17-s + ⋯
L(s)  = 1  − 0.535i·2-s + (−1.55 + 0.896i)3-s + 0.712·4-s + (−1.36 + 0.789i)5-s + (0.480 + 0.832i)6-s + (0.652 + 0.376i)7-s − 0.917i·8-s + (1.10 − 1.91i)9-s + (0.423 + 0.733i)10-s − 1.21·11-s + (−1.10 + 0.639i)12-s + (0.501 − 0.869i)13-s + (0.201 − 0.349i)14-s + (1.41 − 2.45i)15-s + 0.220·16-s + (−0.0426 + 0.0738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0958 + 0.995i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0958 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.0958 + 0.995i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ -0.0958 + 0.995i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.351489 - 0.386959i\)
\(L(\frac12)\) \(\approx\) \(0.351489 - 0.386959i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (7.81e4 + 1.44e4i)T \)
good2 \( 1 + 4.28iT - 64T^{2} \)
3 \( 1 + (41.9 - 24.2i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (171. - 98.7i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-223. - 129. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + 1.62e3T + 1.77e6T^{2} \)
13 \( 1 + (-1.10e3 + 1.90e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (209. - 362. i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-4.69e3 + 2.70e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (9.99e3 + 1.73e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-2.03e4 - 1.17e4i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-7.07e3 - 1.22e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (1.42e4 - 8.21e3i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 6.64e4T + 4.75e9T^{2} \)
47 \( 1 - 3.06e4T + 1.07e10T^{2} \)
53 \( 1 + (1.02e5 + 1.76e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + 3.26e5T + 4.21e10T^{2} \)
61 \( 1 + (2.77e5 + 1.60e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.10e5 + 1.91e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (-1.53e5 - 8.84e4i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (4.70e5 + 2.71e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-3.00e5 + 5.19e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-3.87e5 - 6.71e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (-1.01e6 + 5.86e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 4.17e5T + 8.32e11T^{2} \)
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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

−15.01896012611367562716528998687, −12.41652543317553045181202486495, −11.69947924768048263094870502894, −10.76789183292963744607037279993, −10.44069425720677676793775083090, −7.86212750139262236381763803465, −6.38391581849889717575115676614, −4.83799682600661162047366357498, −3.22702028838690303852190957222, −0.32223682412904537949064935509, 1.33349574776233432371727580568, 4.71219021911629836235067412629, 5.93682675806923851941568992944, 7.49478991892938323382725125917, 7.895173401726108008381148633842, 10.82110675252390143500943384481, 11.63687529098568119295747967254, 12.19054592207428287614102401442, 13.65098405552508567588437013178, 15.67523419278770585169234444593

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