Properties

Degree 2
Conductor 43
Sign $-0.937 + 0.346i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.53 + 3.18i)2-s + (57.5 − 119. i)3-s + (151. − 190. i)4-s + (−620. − 141. i)5-s + 469.·6-s − 2.75e3i·7-s + (1.72e3 + 393. i)8-s + (−6.86e3 − 8.60e3i)9-s + (−501. − 2.19e3i)10-s + (3.91e3 + 4.91e3i)11-s + (−1.40e4 − 2.90e4i)12-s + (−4.45e3 + 1.95e4i)13-s + (8.77e3 − 4.22e3i)14-s + (−5.25e4 + 6.59e4i)15-s + (−1.24e4 − 5.46e4i)16-s + (2.07e4 + 9.08e4i)17-s + ⋯
L(s)  = 1  + (0.0959 + 0.199i)2-s + (0.709 − 1.47i)3-s + (0.592 − 0.743i)4-s + (−0.992 − 0.226i)5-s + 0.361·6-s − 1.14i·7-s + (0.420 + 0.0960i)8-s + (−1.04 − 1.31i)9-s + (−0.0501 − 0.219i)10-s + (0.267 + 0.335i)11-s + (−0.675 − 1.40i)12-s + (−0.156 + 0.683i)13-s + (0.228 − 0.110i)14-s + (−1.03 + 1.30i)15-s + (−0.190 − 0.834i)16-s + (0.248 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.346i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.937 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.937 + 0.346i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.937 + 0.346i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.382343 - 2.13756i\)
\(L(\frac12)\)  \(\approx\)  \(0.382343 - 2.13756i\)
\(L(5)\)   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 \( 1 + (-2.17e6 + 2.63e6i)T \)
good2 \( 1 + (-1.53 - 3.18i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (-57.5 + 119. i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (620. + 141. i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 + 2.75e3iT - 5.76e6T^{2} \)
11 \( 1 + (-3.91e3 - 4.91e3i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (4.45e3 - 1.95e4i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (-2.07e4 - 9.08e4i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (-1.99e5 - 1.59e5i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (2.93e5 + 3.67e5i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (2.55e5 + 5.29e5i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (-5.84e5 + 2.81e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 + 2.13e6iT - 3.51e12T^{2} \)
41 \( 1 + (2.47e6 - 1.19e6i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (3.14e6 - 3.93e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (-7.78e5 - 3.41e6i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (8.37e5 + 3.67e6i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (-1.01e7 + 2.10e7i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (5.77e6 - 7.23e6i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (-2.91e7 - 2.32e7i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (-1.71e7 - 3.92e6i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 + 2.84e7T + 1.51e15T^{2} \)
83 \( 1 + (2.54e7 + 1.22e7i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (-1.95e7 + 4.05e7i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (-6.02e7 - 7.55e7i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−13.99997176095228308378326825739, −12.50531586114077688674826721455, −11.61335229323049296592082284451, −10.03567635213388256989219456294, −8.044535979017100130780075426704, −7.40163585844300750194939417221, −6.29517608324340457367964234454, −3.95278268197887636249025118012, −1.88275596348911375254906747105, −0.73820465884531759347511154648, 2.87197376694987549979974479445, 3.50622544075709752313776924423, 5.12777727163361818620462766304, 7.50002169137364420408207280484, 8.626348917688219460109191525699, 9.789227086919547949480967102878, 11.38464527272907780156002725752, 11.88410589228486329807006509677, 13.73473484925436075840987189701, 15.27271502262965873571057015160

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