Properties

Degree 2
Conductor 43
Sign $0.962 - 0.271i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.911 − 1.89i)2-s + (5.18 − 10.7i)3-s + (156. − 196. i)4-s + (618. + 141. i)5-s − 25.1·6-s + 3.95e3i·7-s + (−1.03e3 − 237. i)8-s + (4.00e3 + 5.01e3i)9-s + (−296. − 1.30e3i)10-s + (−8.25e3 − 1.03e4i)11-s + (−1.30e3 − 2.70e3i)12-s + (409. − 1.79e3i)13-s + (7.48e3 − 3.60e3i)14-s + (4.73e3 − 5.93e3i)15-s + (−1.38e4 − 6.06e4i)16-s + (2.86e4 + 1.25e5i)17-s + ⋯
L(s)  = 1  + (−0.0569 − 0.118i)2-s + (0.0640 − 0.132i)3-s + (0.612 − 0.768i)4-s + (0.990 + 0.226i)5-s − 0.0193·6-s + 1.64i·7-s + (−0.253 − 0.0579i)8-s + (0.609 + 0.764i)9-s + (−0.0296 − 0.130i)10-s + (−0.564 − 0.707i)11-s + (−0.0629 − 0.130i)12-s + (0.0143 − 0.0627i)13-s + (0.194 − 0.0938i)14-s + (0.0934 − 0.117i)15-s + (−0.211 − 0.924i)16-s + (0.342 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.962 - 0.271i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ 0.962 - 0.271i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(2.54293 + 0.351733i\)
\(L(\frac12)\)  \(\approx\)  \(2.54293 + 0.351733i\)
\(L(5)\)   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 \( 1 + (2.37e6 - 2.45e6i)T \)
good2 \( 1 + (0.911 + 1.89i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (-5.18 + 10.7i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (-618. - 141. i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 - 3.95e3iT - 5.76e6T^{2} \)
11 \( 1 + (8.25e3 + 1.03e4i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (-409. + 1.79e3i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (-2.86e4 - 1.25e5i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (-1.45e5 - 1.15e5i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (-4.86e4 - 6.10e4i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (1.72e5 + 3.59e5i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (-1.44e6 + 6.95e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 + 1.81e6iT - 3.51e12T^{2} \)
41 \( 1 + (-2.87e6 + 1.38e6i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (-1.73e5 + 2.17e5i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (1.20e6 + 5.27e6i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (-1.85e5 - 8.11e5i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (-3.63e6 + 7.54e6i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (2.32e7 - 2.91e7i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (1.64e7 + 1.31e7i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (6.53e6 + 1.49e6i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 - 3.09e7T + 1.51e15T^{2} \)
83 \( 1 + (2.27e7 + 1.09e7i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (9.40e6 - 1.95e7i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (1.03e7 + 1.29e7i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−14.32799743693075697314601473085, −13.12258530961475795726341162776, −11.81697574623502858897206816924, −10.49236927315402912203391029283, −9.614403740837379298667724462945, −8.031314882779082609773353372048, −6.05335722758171055249569858298, −5.54743580342100803589406306078, −2.59997724231326504589200535737, −1.62584272200986785856891122568, 1.09299161838975748455915456045, 3.02593155690145962732787316672, 4.69074191094374867677968594420, 6.77912800165329738401652402024, 7.49983847956256926743499552744, 9.426883697883422511397205792292, 10.35003600184669953710514795166, 11.84049356101313087254422991737, 13.14505164750233093677356054373, 13.90651745133541814209316616550

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