Properties

Degree 2
Conductor 43
Sign $-0.933 + 0.359i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.245 + 0.308i)2-s + (4.65 + 5.84i)3-s + (7.08 + 31.0i)4-s + (−95.9 − 46.2i)5-s − 2.94·6-s − 56.9·7-s + (−22.6 − 10.9i)8-s + (41.6 − 182. i)9-s + (37.8 − 18.2i)10-s + (15.4 − 67.7i)11-s + (−148. + 186. i)12-s + (−421. − 202. i)13-s + (13.9 − 17.5i)14-s + (−177. − 776. i)15-s + (−909. + 437. i)16-s + (−1.78e3 + 857. i)17-s + ⋯
L(s)  = 1  + (−0.0434 + 0.0544i)2-s + (0.298 + 0.374i)3-s + (0.221 + 0.970i)4-s + (−1.71 − 0.826i)5-s − 0.0334·6-s − 0.439·7-s + (−0.125 − 0.0603i)8-s + (0.171 − 0.750i)9-s + (0.119 − 0.0576i)10-s + (0.0385 − 0.168i)11-s + (−0.297 + 0.373i)12-s + (−0.691 − 0.332i)13-s + (0.0190 − 0.0239i)14-s + (−0.203 − 0.890i)15-s + (−0.887 + 0.427i)16-s + (−1.49 + 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.933 + 0.359i$
motivic weight  =  \(5\)
character  :  $\chi_{43} (4, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ -0.933 + 0.359i)\)
\(L(3)\)  \(\approx\)  \(0.0109471 - 0.0588478i\)
\(L(\frac12)\)  \(\approx\)  \(0.0109471 - 0.0588478i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-1.18e4 + 2.60e3i)T \)
good2 \( 1 + (0.245 - 0.308i)T + (-7.12 - 31.1i)T^{2} \)
3 \( 1 + (-4.65 - 5.84i)T + (-54.0 + 236. i)T^{2} \)
5 \( 1 + (95.9 + 46.2i)T + (1.94e3 + 2.44e3i)T^{2} \)
7 \( 1 + 56.9T + 1.68e4T^{2} \)
11 \( 1 + (-15.4 + 67.7i)T + (-1.45e5 - 6.98e4i)T^{2} \)
13 \( 1 + (421. + 202. i)T + (2.31e5 + 2.90e5i)T^{2} \)
17 \( 1 + (1.78e3 - 857. i)T + (8.85e5 - 1.11e6i)T^{2} \)
19 \( 1 + (-371. - 1.62e3i)T + (-2.23e6 + 1.07e6i)T^{2} \)
23 \( 1 + (-582. + 2.55e3i)T + (-5.79e6 - 2.79e6i)T^{2} \)
29 \( 1 + (1.91e3 - 2.39e3i)T + (-4.56e6 - 1.99e7i)T^{2} \)
31 \( 1 + (3.75e3 - 4.71e3i)T + (-6.37e6 - 2.79e7i)T^{2} \)
37 \( 1 + 1.08e3T + 6.93e7T^{2} \)
41 \( 1 + (-5.21e3 + 6.54e3i)T + (-2.57e7 - 1.12e8i)T^{2} \)
47 \( 1 + (1.51e3 + 6.63e3i)T + (-2.06e8 + 9.95e7i)T^{2} \)
53 \( 1 + (2.91e4 - 1.40e4i)T + (2.60e8 - 3.26e8i)T^{2} \)
59 \( 1 + (-3.27e4 + 1.57e4i)T + (4.45e8 - 5.58e8i)T^{2} \)
61 \( 1 + (1.67e4 + 2.09e4i)T + (-1.87e8 + 8.23e8i)T^{2} \)
67 \( 1 + (-2.71e3 - 1.19e4i)T + (-1.21e9 + 5.85e8i)T^{2} \)
71 \( 1 + (-7.11e3 - 3.11e4i)T + (-1.62e9 + 7.82e8i)T^{2} \)
73 \( 1 + (2.61e4 + 1.25e4i)T + (1.29e9 + 1.62e9i)T^{2} \)
79 \( 1 - 823.T + 3.07e9T^{2} \)
83 \( 1 + (1.87e4 + 2.35e4i)T + (-8.76e8 + 3.84e9i)T^{2} \)
89 \( 1 + (-3.15e4 - 3.95e4i)T + (-1.24e9 + 5.44e9i)T^{2} \)
97 \( 1 + (1.28e4 - 5.63e4i)T + (-7.73e9 - 3.72e9i)T^{2} \)
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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

−15.80024569627695962203101684324, −14.82314547744100289625243001779, −12.63041866587381246250194795454, −12.41771770775482494424618756521, −11.03534696483293239701200270343, −9.047028634576154560941874857791, −8.223595761617119944868856048044, −6.97442826572639823218747010066, −4.34172949925564732244116471355, −3.41494396496293600544209717819, 0.03025168188454925319640351764, 2.57637885440120963855739506474, 4.58474194141983013202491367631, 6.82530329430166155900720438599, 7.57537856370424728846312198718, 9.375145830390012187344801151345, 10.97371935579524297241183511829, 11.53164776409014090797504620635, 13.22903849285861742785337269668, 14.50114693556452597744302956311

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