Properties

Degree 2
Conductor 43
Sign $0.896 - 0.443i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.4i·2-s + (−34.2 − 19.7i)3-s − 116.·4-s + (−28.7 − 16.6i)5-s + (265. − 459. i)6-s + (−186. + 107. i)7-s − 704. i·8-s + (415. + 720. i)9-s + (223. − 386. i)10-s + 2.28e3·11-s + (3.98e3 + 2.29e3i)12-s + (−344. − 596. i)13-s + (−1.44e3 − 2.50e3i)14-s + (656. + 1.13e3i)15-s + 2.00e3·16-s + (−510. − 883. i)17-s + ⋯
L(s)  = 1  + 1.67i·2-s + (−1.26 − 0.731i)3-s − 1.81·4-s + (−0.230 − 0.132i)5-s + (1.22 − 2.12i)6-s + (−0.543 + 0.313i)7-s − 1.37i·8-s + (0.570 + 0.988i)9-s + (0.223 − 0.386i)10-s + 1.71·11-s + (2.30 + 1.33i)12-s + (−0.156 − 0.271i)13-s + (−0.526 − 0.912i)14-s + (0.194 + 0.336i)15-s + 0.490·16-s + (−0.103 − 0.179i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.896 - 0.443i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.896 - 0.443i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.732490 + 0.171159i\)
\(L(\frac12)\)  \(\approx\)  \(0.732490 + 0.171159i\)
\(L(4)\)   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 \( 1 + (4.12e4 + 6.79e4i)T \)
good2 \( 1 - 13.4iT - 64T^{2} \)
3 \( 1 + (34.2 + 19.7i)T + (364.5 + 631. i)T^{2} \)
5 \( 1 + (28.7 + 16.6i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (186. - 107. i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 - 2.28e3T + 1.77e6T^{2} \)
13 \( 1 + (344. + 596. i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + (510. + 883. i)T + (-1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-461. - 266. i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-4.19e3 + 7.27e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-3.28e4 + 1.89e4i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-1.20e4 + 2.09e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-7.24e4 - 4.18e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 6.00e4T + 4.75e9T^{2} \)
47 \( 1 + 1.84e5T + 1.07e10T^{2} \)
53 \( 1 + (-1.23e4 + 2.14e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + 3.37e5T + 4.21e10T^{2} \)
61 \( 1 + (-6.32e4 + 3.65e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-2.37e5 + 4.11e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + (-1.76e5 + 1.02e5i)T + (6.40e10 - 1.10e11i)T^{2} \)
73 \( 1 + (-1.85e5 + 1.07e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (5.51e4 + 9.55e4i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (1.83e5 - 3.17e5i)T + (-1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + (-1.00e6 - 5.82e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 5.97e5T + 8.32e11T^{2} \)
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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

−15.03064129246375493171632608495, −13.81234030694059236183950481973, −12.49932733182126689503245140456, −11.55093553281382543837288834225, −9.490241733717224203888229254113, −8.024931947085952391110759961144, −6.51807890825023024897681489431, −6.25130085516614514371215424110, −4.62635068497229074795960608920, −0.54889944266192950816339370142, 1.07942562087223403732503570781, 3.53213172621964049449780235903, 4.62738784990759466196827350455, 6.50107022808918768219068985100, 9.242158480932789265591931948433, 10.08005454960811120056099506395, 11.25319834906848320412940258798, 11.72689471875975129019734505818, 12.85856393340192292706016081688, 14.35069382136714754961646653650

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