Properties

Degree 2
Conductor 43
Sign $0.289 - 0.957i$
Motivic weight 7
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.65 + 10.8i)2-s + (35.8 + 5.40i)3-s + (−14.4 + 63.1i)4-s + (369. + 251. i)5-s + (251. + 436. i)6-s + (471. − 816. i)7-s + (790. − 380. i)8-s + (−831. − 256. i)9-s + (463. + 6.18e3i)10-s + (355. + 1.55e3i)11-s + (−859. + 2.18e3i)12-s + (300. − 4.00e3i)13-s + (1.29e4 − 1.95e3i)14-s + (1.18e4 + 1.10e4i)15-s + (1.84e4 + 8.88e3i)16-s + (−4.28e3 + 2.92e3i)17-s + ⋯
L(s)  = 1  + (0.765 + 0.959i)2-s + (0.767 + 0.115i)3-s + (−0.112 + 0.493i)4-s + (1.32 + 0.900i)5-s + (0.476 + 0.824i)6-s + (0.519 − 0.899i)7-s + (0.545 − 0.262i)8-s + (−0.380 − 0.117i)9-s + (0.146 + 1.95i)10-s + (0.0806 + 0.353i)11-s + (−0.143 + 0.365i)12-s + (0.0379 − 0.505i)13-s + (1.26 − 0.190i)14-s + (0.908 + 0.843i)15-s + (1.12 + 0.542i)16-s + (−0.211 + 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.289 - 0.957i$
motivic weight  =  \(7\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :7/2),\ 0.289 - 0.957i)\)
\(L(4)\)  \(\approx\)  \(3.39445 + 2.51855i\)
\(L(\frac12)\)  \(\approx\)  \(3.39445 + 2.51855i\)
\(L(\frac{9}{2})\)   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 \( 1 + (-5.08e5 - 1.16e5i)T \)
good2 \( 1 + (-8.65 - 10.8i)T + (-28.4 + 124. i)T^{2} \)
3 \( 1 + (-35.8 - 5.40i)T + (2.08e3 + 644. i)T^{2} \)
5 \( 1 + (-369. - 251. i)T + (2.85e4 + 7.27e4i)T^{2} \)
7 \( 1 + (-471. + 816. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-355. - 1.55e3i)T + (-1.75e7 + 8.45e6i)T^{2} \)
13 \( 1 + (-300. + 4.00e3i)T + (-6.20e7 - 9.35e6i)T^{2} \)
17 \( 1 + (4.28e3 - 2.92e3i)T + (1.49e8 - 3.81e8i)T^{2} \)
19 \( 1 + (5.09e4 - 1.57e4i)T + (7.38e8 - 5.03e8i)T^{2} \)
23 \( 1 + (3.12e4 - 2.90e4i)T + (2.54e8 - 3.39e9i)T^{2} \)
29 \( 1 + (6.74e4 - 1.01e4i)T + (1.64e10 - 5.08e9i)T^{2} \)
31 \( 1 + (-4.96e4 + 1.26e5i)T + (-2.01e10 - 1.87e10i)T^{2} \)
37 \( 1 + (-8.36e4 - 1.44e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (3.41e5 + 4.28e5i)T + (-4.33e10 + 1.89e11i)T^{2} \)
47 \( 1 + (-8.07e4 + 3.53e5i)T + (-4.56e11 - 2.19e11i)T^{2} \)
53 \( 1 + (1.31e5 + 1.75e6i)T + (-1.16e12 + 1.75e11i)T^{2} \)
59 \( 1 + (-6.83e5 - 3.28e5i)T + (1.55e12 + 1.94e12i)T^{2} \)
61 \( 1 + (4.61e5 + 1.17e6i)T + (-2.30e12 + 2.13e12i)T^{2} \)
67 \( 1 + (-1.13e6 + 3.49e5i)T + (5.00e12 - 3.41e12i)T^{2} \)
71 \( 1 + (8.98e5 + 8.34e5i)T + (6.79e11 + 9.06e12i)T^{2} \)
73 \( 1 + (4.40e5 - 5.88e6i)T + (-1.09e13 - 1.64e12i)T^{2} \)
79 \( 1 + (1.93e6 - 3.34e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-9.27e6 - 1.39e6i)T + (2.59e13 + 7.99e12i)T^{2} \)
89 \( 1 + (8.68e6 + 1.30e6i)T + (4.22e13 + 1.30e13i)T^{2} \)
97 \( 1 + (-2.94e6 - 1.28e7i)T + (-7.27e13 + 3.50e13i)T^{2} \)
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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

−14.56919329945557801793864867485, −13.92657960935943938475108373197, −13.11875700167875178108723359297, −10.79402105883509228787289645466, −9.867024258153633471067020876989, −8.082820919758820753551780577766, −6.75503415265258297244002644730, −5.69443040289524494995724606991, −3.93505675540059073547193735546, −2.05431824243699175430128920950, 1.79616493987745697404738605275, 2.55448199886631000518556590955, 4.57381343393747474329297420762, 5.85336989056021961154880377343, 8.394932383407624570004931541337, 9.167596383354694299222395313238, 10.81060915309278113274119782114, 12.11278129504406715679946721756, 13.11993137959931198778436382174, 13.86708620766418968144269093575

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