Properties

Degree 2
Conductor 43
Sign $0.586 - 0.809i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 26.2·2-s + (−64.5 + 111. i)3-s + 178.·4-s + (−856. + 1.48e3i)5-s + (1.69e3 − 2.93e3i)6-s + (−4.32e3 − 7.49e3i)7-s + 8.76e3·8-s + (1.50e3 + 2.61e3i)9-s + (2.25e4 − 3.89e4i)10-s + 4.66e3·11-s + (−1.15e4 + 1.99e4i)12-s + (−9.88e4 − 1.71e5i)13-s + (1.13e5 + 1.96e5i)14-s + (−1.10e5 − 1.91e5i)15-s − 3.21e5·16-s + (−3.49e4 − 6.05e4i)17-s + ⋯
L(s)  = 1  − 1.16·2-s + (−0.460 + 0.796i)3-s + 0.348·4-s + (−0.612 + 1.06i)5-s + (0.534 − 0.925i)6-s + (−0.680 − 1.17i)7-s + 0.756·8-s + (0.0766 + 0.132i)9-s + (0.711 − 1.23i)10-s + 0.0960·11-s + (−0.160 + 0.277i)12-s + (−0.959 − 1.66i)13-s + (0.790 + 1.36i)14-s + (−0.564 − 0.976i)15-s − 1.22·16-s + (−0.101 − 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.586 - 0.809i$
motivic weight  =  \(9\)
character  :  $\chi_{43} (6, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ 0.586 - 0.809i)\)
\(L(5)\)  \(\approx\)  \(0.369260 + 0.188522i\)
\(L(\frac12)\)  \(\approx\)  \(0.369260 + 0.188522i\)
\(L(\frac{11}{2})\)   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 \( 1 + (-2.58e6 - 2.22e7i)T \)
good2 \( 1 + 26.2T + 512T^{2} \)
3 \( 1 + (64.5 - 111. i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (856. - 1.48e3i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 + (4.32e3 + 7.49e3i)T + (-2.01e7 + 3.49e7i)T^{2} \)
11 \( 1 - 4.66e3T + 2.35e9T^{2} \)
13 \( 1 + (9.88e4 + 1.71e5i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + (3.49e4 + 6.05e4i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-7.08e4 + 1.22e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-3.83e5 + 6.64e5i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + (-3.46e6 - 5.99e6i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 + (1.22e6 - 2.11e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (-6.26e6 + 1.08e7i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 1.39e6T + 3.27e14T^{2} \)
47 \( 1 + 2.95e7T + 1.11e15T^{2} \)
53 \( 1 + (-8.91e5 + 1.54e6i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 + 1.36e8T + 8.66e15T^{2} \)
61 \( 1 + (7.89e7 + 1.36e8i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (4.47e7 - 7.75e7i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + (-1.97e7 - 3.41e7i)T + (-2.29e16 + 3.97e16i)T^{2} \)
73 \( 1 + (-1.60e8 - 2.77e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-2.12e8 - 3.68e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 + (-1.83e8 + 3.17e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (-1.60e8 + 2.77e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 - 1.21e9T + 7.60e17T^{2} \)
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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

−14.40883286530219473973433099357, −12.90270381840068653709335974954, −10.86422834631114571640818412147, −10.55322658737533100447609405717, −9.648309086176566873542571058143, −7.80271433620671498064192540836, −6.99032041354668412479433507518, −4.74915008559904924385282367770, −3.21797654897575870921807578109, −0.55141699961892569122996901475, 0.49206487258117608507835874590, 1.86468891237172150594007669382, 4.55015464151973171447970172075, 6.37465464502349089703507185133, 7.70603624033012705894916038269, 8.969100032073800977172826587353, 9.614653216091239977057062047503, 11.75818861973033912982784017911, 12.21570170871717174983896772382, 13.45863670769612361845665013738

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