Properties

Degree 2
Conductor 43
Sign $-0.0958 - 0.995i$
Motivic weight 6
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

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.0958 - 0.995i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.0958 - 0.995i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.351489 + 0.386959i\)
\(L(\frac12)\)  \(\approx\)  \(0.351489 + 0.386959i\)
\(L(4)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]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 \( 1 + (7.81e4 - 1.44e4i)T \)
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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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