Properties

Degree 2
Conductor 43
Sign $-0.924 + 0.380i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.84i·2-s + (36.4 − 21.0i)3-s + 2.53·4-s + (−116. + 67.3i)5-s + (−165. − 286. i)6-s + (−385. − 222. i)7-s − 521. i·8-s + (522. − 905. i)9-s + (528. + 914. i)10-s − 327.·11-s + (92.4 − 53.3i)12-s + (1.05e3 − 1.82e3i)13-s + (−1.74e3 + 3.01e3i)14-s + (−2.83e3 + 4.91e3i)15-s − 3.92e3·16-s + (2.72e3 − 4.72e3i)17-s + ⋯
L(s)  = 1  − 0.980i·2-s + (1.35 − 0.780i)3-s + 0.0395·4-s + (−0.933 + 0.538i)5-s + (−0.764 − 1.32i)6-s + (−1.12 − 0.648i)7-s − 1.01i·8-s + (0.717 − 1.24i)9-s + (0.528 + 0.914i)10-s − 0.246·11-s + (0.0534 − 0.0308i)12-s + (0.478 − 0.829i)13-s + (−0.635 + 1.10i)14-s + (−0.840 + 1.45i)15-s − 0.958·16-s + (0.555 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.924 + 0.380i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.924 + 0.380i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.419665 - 2.12533i\)
\(L(\frac12)\)  \(\approx\)  \(0.419665 - 2.12533i\)
\(L(4)\)   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 \( 1 + (2.37e4 + 7.58e4i)T \)
good2 \( 1 + 7.84iT - 64T^{2} \)
3 \( 1 + (-36.4 + 21.0i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (116. - 67.3i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (385. + 222. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + 327.T + 1.77e6T^{2} \)
13 \( 1 + (-1.05e3 + 1.82e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (-2.72e3 + 4.72e3i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-3.87e3 + 2.23e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-8.46e3 - 1.46e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-1.25e4 - 7.26e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-2.57e4 - 4.46e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-7.08e4 + 4.08e4i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 1.83e4T + 4.75e9T^{2} \)
47 \( 1 + 5.58e4T + 1.07e10T^{2} \)
53 \( 1 + (-8.64e4 - 1.49e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 - 2.42e4T + 4.21e10T^{2} \)
61 \( 1 + (1.03e5 + 5.95e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (2.46e5 + 4.26e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (-3.18e5 - 1.83e5i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (5.17e5 + 2.98e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-7.99e4 + 1.38e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-5.67e4 - 9.82e4i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (2.55e5 - 1.47e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.59e6T + 8.32e11T^{2} \)
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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

−13.77342092427460698231005127687, −13.03493833049263700309840127145, −11.92358914182054728026568550690, −10.58438372686423308199330035148, −9.364169117197652092028582787331, −7.65926182945728771218074012915, −6.94918783837678792020895754744, −3.36181653221028918175146634340, −3.03537307960645609297362723442, −0.906726618899109818000942011911, 2.82534477523761897425353197769, 4.31338087125951107165884684123, 6.27204861122290964467652354671, 7.982891857482589199644902300198, 8.641022050630101272394507964355, 9.874319393795798068630096535146, 11.73289607358936849339021396025, 13.19015704023922902506139267246, 14.59978594083005716316814676089, 15.29510363537159266783889557876

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