Properties

Degree 2
Conductor 43
Sign $-0.998 + 0.0525i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.79 − 9.95i)2-s + (−6.97 + 14.4i)3-s + (−36.1 + 45.3i)4-s + (117. + 26.7i)5-s + 177.·6-s − 466. i·7-s + (−64.6 − 14.7i)8-s + (293. + 367. i)9-s + (−295. − 1.29e3i)10-s + (−1.08e3 − 1.36e3i)11-s + (−404. − 839. i)12-s + (211. − 927. i)13-s + (−4.64e3 + 2.23e3i)14-s + (−1.20e3 + 1.50e3i)15-s + (988. + 4.33e3i)16-s + (1.08e3 + 4.75e3i)17-s + ⋯
L(s)  = 1  + (−0.599 − 1.24i)2-s + (−0.258 + 0.536i)3-s + (−0.565 + 0.708i)4-s + (0.936 + 0.213i)5-s + 0.821·6-s − 1.36i·7-s + (−0.126 − 0.0287i)8-s + (0.402 + 0.504i)9-s + (−0.295 − 1.29i)10-s + (−0.815 − 1.02i)11-s + (−0.234 − 0.486i)12-s + (0.0963 − 0.422i)13-s + (−1.69 + 0.814i)14-s + (−0.356 + 0.447i)15-s + (0.241 + 1.05i)16-s + (0.220 + 0.967i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0525i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.998 + 0.0525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.998 + 0.0525i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.998 + 0.0525i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.0224124 - 0.852518i\)
\(L(\frac12)\)  \(\approx\)  \(0.0224124 - 0.852518i\)
\(L(4)\)   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 \( 1 + (6.64e4 - 4.35e4i)T \)
good2 \( 1 + (4.79 + 9.95i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (6.97 - 14.4i)T + (-454. - 569. i)T^{2} \)
5 \( 1 + (-117. - 26.7i)T + (1.40e4 + 6.77e3i)T^{2} \)
7 \( 1 + 466. iT - 1.17e5T^{2} \)
11 \( 1 + (1.08e3 + 1.36e3i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (-211. + 927. i)T + (-4.34e6 - 2.09e6i)T^{2} \)
17 \( 1 + (-1.08e3 - 4.75e3i)T + (-2.17e7 + 1.04e7i)T^{2} \)
19 \( 1 + (9.69e3 + 7.73e3i)T + (1.04e7 + 4.58e7i)T^{2} \)
23 \( 1 + (6.83e3 + 8.57e3i)T + (-3.29e7 + 1.44e8i)T^{2} \)
29 \( 1 + (1.74e4 + 3.61e4i)T + (-3.70e8 + 4.65e8i)T^{2} \)
31 \( 1 + (-1.48e4 + 7.17e3i)T + (5.53e8 - 6.93e8i)T^{2} \)
37 \( 1 - 3.04e4iT - 2.56e9T^{2} \)
41 \( 1 + (-1.08e5 + 5.21e4i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (-4.68e4 + 5.87e4i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (-3.27e4 - 1.43e5i)T + (-1.99e10 + 9.61e9i)T^{2} \)
59 \( 1 + (1.90e4 + 8.36e4i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (-4.50e4 + 9.35e4i)T + (-3.21e10 - 4.02e10i)T^{2} \)
67 \( 1 + (-9.68e4 + 1.21e5i)T + (-2.01e10 - 8.81e10i)T^{2} \)
71 \( 1 + (-4.13e5 - 3.29e5i)T + (2.85e10 + 1.24e11i)T^{2} \)
73 \( 1 + (4.44e5 + 1.01e5i)T + (1.36e11 + 6.56e10i)T^{2} \)
79 \( 1 - 2.67e5T + 2.43e11T^{2} \)
83 \( 1 + (-7.07e5 - 3.40e5i)T + (2.03e11 + 2.55e11i)T^{2} \)
89 \( 1 + (2.65e5 - 5.52e5i)T + (-3.09e11 - 3.88e11i)T^{2} \)
97 \( 1 + (3.22e5 + 4.04e5i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−13.64893288659267095741708933084, −12.97780156569352001461923054132, −10.97944384930843847028891244105, −10.57259327238483213177372131111, −9.828325534666045132748884424219, −8.159293312584636052115633532502, −6.08779331094203356925281922152, −4.08850915196314947112655623764, −2.27709094101204376167683283946, −0.48101378166270458839898570086, 2.01879451408589451010899806793, 5.40057287704333202356763818296, 6.29602125565981695875584621732, 7.55042810530175615107879904795, 8.975425699524819624768726360080, 9.826339195010626657145664425954, 12.04254030201848771145615578970, 12.87110261542237530107942504494, 14.50823437427007541952337461166, 15.41712936917171420436123306851

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