Properties

Degree 2
Conductor 43
Sign $0.984 + 0.175i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.66 + 7.60i)2-s + (−55.7 + 115. i)3-s + (115. − 144. i)4-s + (474. + 108. i)5-s − 1.08e3·6-s − 3.60e3i·7-s + (3.62e3 + 827. i)8-s + (−6.19e3 − 7.76e3i)9-s + (913. + 4.00e3i)10-s + (−1.35e4 − 1.70e4i)11-s + (1.02e4 + 2.13e4i)12-s + (8.86e3 − 3.88e4i)13-s + (2.74e4 − 1.32e4i)14-s + (−3.89e4 + 4.88e4i)15-s + (−3.54e3 − 1.55e4i)16-s + (−7.75e3 − 3.39e4i)17-s + ⋯
L(s)  = 1  + (0.228 + 0.475i)2-s + (−0.687 + 1.42i)3-s + (0.450 − 0.564i)4-s + (0.759 + 0.173i)5-s − 0.836·6-s − 1.50i·7-s + (0.885 + 0.202i)8-s + (−0.943 − 1.18i)9-s + (0.0913 + 0.400i)10-s + (−0.928 − 1.16i)11-s + (0.496 + 1.03i)12-s + (0.310 − 1.35i)13-s + (0.714 − 0.343i)14-s + (−0.769 + 0.964i)15-s + (−0.0540 − 0.236i)16-s + (−0.0927 − 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.984 + 0.175i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.984 + 0.175i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(1.90681 - 0.168506i\)
\(L(\frac12)\)  \(\approx\)  \(1.90681 - 0.168506i\)
\(L(5)\)   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 \( 1 + (3.20e6 - 1.17e6i)T \)
good2 \( 1 + (-3.66 - 7.60i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (55.7 - 115. i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (-474. - 108. i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 + 3.60e3iT - 5.76e6T^{2} \)
11 \( 1 + (1.35e4 + 1.70e4i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (-8.86e3 + 3.88e4i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (7.75e3 + 3.39e4i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (-1.79e5 - 1.43e5i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (-5.24e4 - 6.57e4i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (4.97e4 + 1.03e5i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (9.10e5 - 4.38e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 + 3.94e5iT - 3.51e12T^{2} \)
41 \( 1 + (-2.30e6 + 1.11e6i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (-2.95e6 + 3.70e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (-1.65e6 - 7.25e6i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (2.99e6 + 1.31e7i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (2.83e6 - 5.89e6i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (5.81e6 - 7.28e6i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (-3.24e7 - 2.58e7i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (-2.25e7 - 5.14e6i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 - 4.33e7T + 1.51e15T^{2} \)
83 \( 1 + (3.87e7 + 1.86e7i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (-4.93e7 + 1.02e8i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (-5.39e7 - 6.76e7i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−14.26129792671029662154903569808, −13.49000470806697505762766627584, −11.13420802819660651157800969955, −10.48002612816333389468478265924, −9.899907879202970013123515737330, −7.62233319987235052653212327454, −5.85939794251944053417123113591, −5.24166494078531838947628706113, −3.49337586637317469124513525691, −0.73070371550460577242035841863, 1.73566451122969652866090516303, 2.39252651132649343533320157207, 5.21495467714945310226865063093, 6.52727251713684554480772008428, 7.63041581690195250944012432818, 9.304191696016910704498644361325, 11.24579606411409507666452303098, 12.06081294323830451514204066352, 12.80323667211723975669002796090, 13.60803929683539122538553325799

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