Properties

Degree 2
Conductor 43
Sign $-0.946 - 0.324i$
Motivic weight 8
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.51 − 11.4i)2-s + (38.3 − 79.6i)3-s + (58.8 − 73.7i)4-s + (121. + 27.6i)5-s − 1.12e3·6-s − 1.96e3i·7-s + (−4.34e3 − 991. i)8-s + (−777. − 975. i)9-s + (−351. − 1.54e3i)10-s + (−1.85e3 − 2.32e3i)11-s + (−3.61e3 − 7.50e3i)12-s + (1.14e4 − 5.03e4i)13-s + (−2.25e4 + 1.08e4i)14-s + (6.84e3 − 8.58e3i)15-s + (7.23e3 + 3.16e4i)16-s + (2.35e4 + 1.03e5i)17-s + ⋯
L(s)  = 1  + (−0.344 − 0.716i)2-s + (0.473 − 0.982i)3-s + (0.229 − 0.288i)4-s + (0.193 + 0.0442i)5-s − 0.866·6-s − 0.818i·7-s + (−1.06 − 0.241i)8-s + (−0.118 − 0.148i)9-s + (−0.0351 − 0.154i)10-s + (−0.126 − 0.159i)11-s + (−0.174 − 0.362i)12-s + (0.402 − 1.76i)13-s + (−0.586 + 0.282i)14-s + (0.135 − 0.169i)15-s + (0.110 + 0.483i)16-s + (0.281 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.324i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.946 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.946 - 0.324i$
motivic weight  =  \(8\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :4),\ -0.946 - 0.324i)\)
\(L(\frac{9}{2})\)  \(\approx\)  \(0.283162 + 1.69992i\)
\(L(\frac12)\)  \(\approx\)  \(0.283162 + 1.69992i\)
\(L(5)\)   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 \( 1 + (-3.35e6 + 6.71e5i)T \)
good2 \( 1 + (5.51 + 11.4i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (-38.3 + 79.6i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (-121. - 27.6i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 + 1.96e3iT - 5.76e6T^{2} \)
11 \( 1 + (1.85e3 + 2.32e3i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (-1.14e4 + 5.03e4i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (-2.35e4 - 1.03e5i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (6.13e4 + 4.89e4i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (-1.32e5 - 1.65e5i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (-1.29e5 - 2.69e5i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (1.48e6 - 7.14e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 - 2.13e6iT - 3.51e12T^{2} \)
41 \( 1 + (-2.75e6 + 1.32e6i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (-1.93e6 + 2.42e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (2.57e6 + 1.12e7i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (2.58e5 + 1.13e6i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (-5.69e6 + 1.18e7i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (1.17e7 - 1.47e7i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (2.35e7 + 1.87e7i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (-4.33e6 - 9.89e5i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 + 9.41e4T + 1.51e15T^{2} \)
83 \( 1 + (3.19e7 + 1.53e7i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (-4.11e7 + 8.55e7i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (7.77e7 + 9.74e7i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−13.24876824341803102342428564822, −12.56578260684463936675469417826, −10.88400859160255012039898248462, −10.21960365875538074441621110989, −8.496833223780813799793554714740, −7.29225464420978223337188797351, −5.84912666188576725979508593358, −3.28573027867573513572997328756, −1.79909383164014931016721692411, −0.68091524629526419518902784881, 2.46583269609135656714285021172, 4.12222411419147813562808338860, 5.91422780151193094035771232549, 7.36079214982937356638960615000, 9.018194352206818327362030742508, 9.334362010672229267492106982820, 11.23580361014592359028440305668, 12.39735871028112895450863788481, 14.18863418264708107411710562262, 15.09339653715765361535876643461

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