Properties

Degree 2
Conductor 43
Sign $-0.492 - 0.870i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 58.5i·2-s + (358. − 207. i)3-s − 2.40e3·4-s + (4.07e3 − 2.35e3i)5-s + (−1.21e4 − 2.10e4i)6-s + (−2.29e4 − 1.32e4i)7-s + 8.08e4i·8-s + (5.63e4 − 9.75e4i)9-s + (−1.37e5 − 2.38e5i)10-s + 2.65e5·11-s + (−8.62e5 + 4.98e5i)12-s + (−1.37e4 + 2.37e4i)13-s + (−7.76e5 + 1.34e6i)14-s + (9.73e5 − 1.68e6i)15-s + 2.27e6·16-s + (1.52e5 − 2.63e5i)17-s + ⋯
L(s)  = 1  − 1.82i·2-s + (1.47 − 0.852i)3-s − 2.34·4-s + (1.30 − 0.752i)5-s + (−1.56 − 2.70i)6-s + (−1.36 − 0.789i)7-s + 2.46i·8-s + (0.953 − 1.65i)9-s + (−1.37 − 2.38i)10-s + 1.65·11-s + (−3.46 + 2.00i)12-s + (−0.0369 + 0.0639i)13-s + (−1.44 + 2.50i)14-s + (1.28 − 2.22i)15-s + 2.16·16-s + (0.107 − 0.185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.492 - 0.870i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.492 - 0.870i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(1.62979 + 2.79621i\)
\(L(\frac12)\)  \(\approx\)  \(1.62979 + 2.79621i\)
\(L(6)\)   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 \( 1 + (-1.33e8 + 6.11e7i)T \)
good2 \( 1 + 58.5iT - 1.02e3T^{2} \)
3 \( 1 + (-358. + 207. i)T + (2.95e4 - 5.11e4i)T^{2} \)
5 \( 1 + (-4.07e3 + 2.35e3i)T + (4.88e6 - 8.45e6i)T^{2} \)
7 \( 1 + (2.29e4 + 1.32e4i)T + (1.41e8 + 2.44e8i)T^{2} \)
11 \( 1 - 2.65e5T + 2.59e10T^{2} \)
13 \( 1 + (1.37e4 - 2.37e4i)T + (-6.89e10 - 1.19e11i)T^{2} \)
17 \( 1 + (-1.52e5 + 2.63e5i)T + (-1.00e12 - 1.74e12i)T^{2} \)
19 \( 1 + (1.96e6 - 1.13e6i)T + (3.06e12 - 5.30e12i)T^{2} \)
23 \( 1 + (-5.68e5 - 9.84e5i)T + (-2.07e13 + 3.58e13i)T^{2} \)
29 \( 1 + (-9.21e6 - 5.31e6i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (-2.30e7 - 3.99e7i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 + (-3.09e7 + 1.78e7i)T + (2.40e15 - 4.16e15i)T^{2} \)
41 \( 1 + 2.78e7T + 1.34e16T^{2} \)
47 \( 1 + 1.13e8T + 5.25e16T^{2} \)
53 \( 1 + (-1.05e8 - 1.82e8i)T + (-8.74e16 + 1.51e17i)T^{2} \)
59 \( 1 - 4.39e8T + 5.11e17T^{2} \)
61 \( 1 + (3.22e8 + 1.86e8i)T + (3.56e17 + 6.17e17i)T^{2} \)
67 \( 1 + (4.08e8 + 7.06e8i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 + (-1.79e9 - 1.03e9i)T + (1.62e18 + 2.81e18i)T^{2} \)
73 \( 1 + (2.67e9 + 1.54e9i)T + (2.14e18 + 3.72e18i)T^{2} \)
79 \( 1 + (-1.38e9 + 2.40e9i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (6.17e8 + 1.06e9i)T + (-7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 + (8.45e9 - 4.88e9i)T + (1.55e19 - 2.70e19i)T^{2} \)
97 \( 1 + 1.51e10T + 7.37e19T^{2} \)
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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

−12.90554127964072447040845259862, −12.29150182078415159941150799378, −10.17048996888357383177267133370, −9.369952897660564854177885863378, −8.772358871264612991489164923112, −6.58209221018678447266384171386, −4.03249584422877555378177366748, −2.94760799811088278588332644218, −1.68197215820138649497688705222, −0.951894492427379440201315051131, 2.65431531592019563552247038241, 4.10011845424400064692961755408, 6.00762255514868610405413499690, 6.73192719365835132873460658386, 8.535254433206866858666121974383, 9.435051430856475045190173040098, 9.822645121302760538323567841533, 13.12067583055191667535181893762, 13.95531326810911888920009768544, 14.75009917035807973937409885958

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