Properties

Degree 2
Conductor 43
Sign $-0.902 + 0.430i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.62 − 11.6i)2-s + (−22.1 + 46.0i)3-s + (−64.8 + 81.2i)4-s + (43.4 + 9.92i)5-s + 661.·6-s + 306. i·7-s + (505. + 115. i)8-s + (−1.17e3 − 1.46e3i)9-s + (−128. − 563. i)10-s + (−52.9 − 66.4i)11-s + (−2.30e3 − 4.78e3i)12-s + (−263. + 1.15e3i)13-s + (3.58e3 − 1.72e3i)14-s + (−1.41e3 + 1.78e3i)15-s + (−13.2 − 58.2i)16-s + (−1.86e3 − 8.16e3i)17-s + ⋯
L(s)  = 1  + (−0.702 − 1.45i)2-s + (−0.820 + 1.70i)3-s + (−1.01 + 1.27i)4-s + (0.347 + 0.0793i)5-s + 3.06·6-s + 0.894i·7-s + (0.986 + 0.225i)8-s + (−1.60 − 2.01i)9-s + (−0.128 − 0.563i)10-s + (−0.0397 − 0.0498i)11-s + (−1.33 − 2.76i)12-s + (−0.119 + 0.525i)13-s + (1.30 − 0.628i)14-s + (−0.420 + 0.527i)15-s + (−0.00324 − 0.0142i)16-s + (−0.379 − 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.902 + 0.430i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.902 + 0.430i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.0357135 - 0.157947i\)
\(L(\frac12)\)  \(\approx\)  \(0.0357135 - 0.157947i\)
\(L(4)\)   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 \( 1 + (4.47e4 - 6.56e4i)T \)
good2 \( 1 + (5.62 + 11.6i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (22.1 - 46.0i)T + (-454. - 569. i)T^{2} \)
5 \( 1 + (-43.4 - 9.92i)T + (1.40e4 + 6.77e3i)T^{2} \)
7 \( 1 - 306. iT - 1.17e5T^{2} \)
11 \( 1 + (52.9 + 66.4i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (263. - 1.15e3i)T + (-4.34e6 - 2.09e6i)T^{2} \)
17 \( 1 + (1.86e3 + 8.16e3i)T + (-2.17e7 + 1.04e7i)T^{2} \)
19 \( 1 + (-5.00e3 - 3.99e3i)T + (1.04e7 + 4.58e7i)T^{2} \)
23 \( 1 + (1.23e4 + 1.55e4i)T + (-3.29e7 + 1.44e8i)T^{2} \)
29 \( 1 + (-3.10e3 - 6.45e3i)T + (-3.70e8 + 4.65e8i)T^{2} \)
31 \( 1 + (-8.27e3 + 3.98e3i)T + (5.53e8 - 6.93e8i)T^{2} \)
37 \( 1 + 1.00e5iT - 2.56e9T^{2} \)
41 \( 1 + (1.14e4 - 5.53e3i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (-4.98e4 + 6.24e4i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (-4.91e4 - 2.15e5i)T + (-1.99e10 + 9.61e9i)T^{2} \)
59 \( 1 + (2.27e4 + 9.96e4i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (4.85e4 - 1.00e5i)T + (-3.21e10 - 4.02e10i)T^{2} \)
67 \( 1 + (9.47e4 - 1.18e5i)T + (-2.01e10 - 8.81e10i)T^{2} \)
71 \( 1 + (-2.14e4 - 1.70e4i)T + (2.85e10 + 1.24e11i)T^{2} \)
73 \( 1 + (5.29e5 + 1.20e5i)T + (1.36e11 + 6.56e10i)T^{2} \)
79 \( 1 + 1.69e5T + 2.43e11T^{2} \)
83 \( 1 + (5.86e5 + 2.82e5i)T + (2.03e11 + 2.55e11i)T^{2} \)
89 \( 1 + (2.97e4 - 6.17e4i)T + (-3.09e11 - 3.88e11i)T^{2} \)
97 \( 1 + (5.27e5 + 6.61e5i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−14.23451964494012748258596250880, −12.11358097146695695756573899134, −11.59984744931964524744341261035, −10.45028140889554748717596612428, −9.615737894403496549509139296574, −8.894205159506992311668804831435, −5.78554196815096608601439141561, −4.27044648788460394384320051592, −2.66648812779326468007547639866, −0.11353299410460106951526162169, 1.34472819395331130865508390806, 5.50560665528796531617674948063, 6.44983529711896180467825744041, 7.45216457426433126485648125166, 8.248951469212196751672489822855, 10.17788344756297607680734799437, 11.74715075132977885623104076647, 13.24357895250071383218068320930, 13.86439673764613093342519758773, 15.44406047572362554727256603599

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