Properties

Degree 2
Conductor 43
Sign $0.970 - 0.240i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.94 + 7.45i)2-s + (86.0 + 12.9i)3-s + (8.24 − 36.1i)4-s + (−256. − 174. i)5-s + (415. + 718. i)6-s + (380. − 658. i)7-s + (1.41e3 − 682. i)8-s + (5.15e3 + 1.58e3i)9-s + (−220. − 2.94e3i)10-s + (1.25e3 + 5.51e3i)11-s + (1.17e3 − 3.00e3i)12-s + (551. − 7.35e3i)13-s + (7.17e3 − 1.08e3i)14-s + (−1.97e4 − 1.83e4i)15-s + (9.25e3 + 4.45e3i)16-s + (−2.98e4 + 2.03e4i)17-s + ⋯
L(s)  = 1  + (0.525 + 0.659i)2-s + (1.84 + 0.277i)3-s + (0.0644 − 0.282i)4-s + (−0.916 − 0.624i)5-s + (0.784 + 1.35i)6-s + (0.419 − 0.725i)7-s + (0.979 − 0.471i)8-s + (2.35 + 0.726i)9-s + (−0.0698 − 0.932i)10-s + (0.284 + 1.24i)11-s + (0.196 − 0.501i)12-s + (0.0696 − 0.928i)13-s + (0.698 − 0.105i)14-s + (−1.51 − 1.40i)15-s + (0.564 + 0.271i)16-s + (−1.47 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.970 - 0.240i$
motivic weight  =  \(7\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ 0.970 - 0.240i)\)
\(L(4)\)  \(\approx\)  \(4.10665 + 0.500289i\)
\(L(\frac12)\)  \(\approx\)  \(4.10665 + 0.500289i\)
\(L(\frac{9}{2})\)   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 \( 1 + (-1.59e5 - 4.96e5i)T \)
good2 \( 1 + (-5.94 - 7.45i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (-86.0 - 12.9i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (256. + 174. i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (-380. + 658. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-1.25e3 - 5.51e3i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (-551. + 7.35e3i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (2.98e4 - 2.03e4i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (-3.54e4 + 1.09e4i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (4.11e4 - 3.82e4i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (9.52e4 - 1.43e4i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (6.08e4 - 1.55e5i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (1.49e5 + 2.58e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (-4.34e4 - 5.44e4i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (-3.36e4 + 1.47e5i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (-1.13e4 - 1.51e5i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (2.46e6 + 1.18e6i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (-3.17e5 - 8.07e5i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (1.16e5 - 3.58e4i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (3.33e5 + 3.09e5i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (3.16e4 - 4.22e5i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (-2.06e6 + 3.57e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (3.82e6 + 5.76e5i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (-9.71e6 - 1.46e6i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (-4.77e5 - 2.09e6i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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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

−14.65474141242050791484123522925, −13.66965247451343600852758503782, −12.73322640311064011152720653899, −10.59583738456546466900911524646, −9.320539175561721699724804696129, −7.934340306871100705526006863015, −7.26662907055985549718739889677, −4.66884321032684555063839402058, −3.85201267108950479726490862970, −1.61419640555556558597856451579, 2.08141979468708092781099535910, 3.14638557658055430940245921247, 4.11755416948463250082384557427, 7.16161974890451366407169338734, 8.187232432258449509229526840357, 9.142399217219946423965983651519, 11.22073473159691324862684526153, 12.01655616502994208532584088310, 13.57357998461358196642699871496, 14.04423993566951841102407431357

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