Properties

Degree 2
Conductor 43
Sign $0.304 - 0.952i$
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 + (−21.3 + 44.3i)3-s + (115. − 144. i)4-s + (−661. − 151. i)5-s − 414.·6-s − 1.49e3i·7-s + (3.62e3 + 827. i)8-s + (2.58e3 + 3.23e3i)9-s + (−1.27e3 − 5.58e3i)10-s + (1.18e4 + 1.48e4i)11-s + (3.94e3 + 8.18e3i)12-s + (886. − 3.88e3i)13-s + (1.13e4 − 5.47e3i)14-s + (2.08e4 − 2.61e4i)15-s + (−3.54e3 − 1.55e4i)16-s + (1.37e4 + 6.01e4i)17-s + ⋯
L(s)  = 1  + (0.228 + 0.475i)2-s + (−0.263 + 0.547i)3-s + (0.450 − 0.564i)4-s + (−1.05 − 0.241i)5-s − 0.320·6-s − 0.623i·7-s + (0.885 + 0.202i)8-s + (0.393 + 0.493i)9-s + (−0.127 − 0.558i)10-s + (0.808 + 1.01i)11-s + (0.190 + 0.394i)12-s + (0.0310 − 0.135i)13-s + (0.296 − 0.142i)14-s + (0.411 − 0.515i)15-s + (−0.0541 − 0.237i)16-s + (0.164 + 0.720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.304 - 0.952i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.304 - 0.952i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(1.58255 + 1.15564i\)
\(L(\frac12)\)  \(\approx\)  \(1.58255 + 1.15564i\)
\(L(5)\)   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 \( 1 + (-7.41e5 - 3.33e6i)T \)
good2 \( 1 + (-3.66 - 7.60i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (21.3 - 44.3i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (661. + 151. i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 + 1.49e3iT - 5.76e6T^{2} \)
11 \( 1 + (-1.18e4 - 1.48e4i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (-886. + 3.88e3i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (-1.37e4 - 6.01e4i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (-4.23e3 - 3.37e3i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (-3.04e5 - 3.81e5i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (-2.97e5 - 6.18e5i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (-9.78e5 + 4.71e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 - 1.33e6iT - 3.51e12T^{2} \)
41 \( 1 + (1.03e6 - 4.96e5i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (-4.02e6 + 5.04e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (2.36e6 + 1.03e7i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (1.70e6 + 7.45e6i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (5.65e5 - 1.17e6i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (7.28e6 - 9.13e6i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (8.44e6 + 6.73e6i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (1.13e7 + 2.60e6i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 - 2.23e7T + 1.51e15T^{2} \)
83 \( 1 + (-8.29e7 - 3.99e7i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (1.77e7 - 3.68e7i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (5.45e7 + 6.84e7i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−14.89333647805585429194756337687, −13.43564116611186365986033219599, −11.89364571577677654044072595595, −10.82982685000081771767691426827, −9.794383507213059705416810189736, −7.82106342584063826843099850400, −6.83272940222505878659359357378, −5.02997262783108951496021870340, −4.01670754982591609418679019876, −1.33186326667194408778014161983, 0.846276149019290657528595160722, 2.83361560352378579693853632645, 4.12417079047560098203725395889, 6.41241122475640225891463805791, 7.48729788212962786189671263620, 8.837523592375683646763058783714, 10.88059080509271365085636638398, 11.91989916025033217025104217660, 12.26108486508335455765452813834, 13.72750627423895440787199899013

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