Properties

Degree 2
Conductor 43
Sign $0.915 + 0.402i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.64i·2-s + (−0.554 + 0.320i)3-s + 57.0·4-s + (14.6 − 8.48i)5-s + (0.846 + 1.46i)6-s + (131. + 76.0i)7-s − 320. i·8-s + (−364. + 630. i)9-s + (−22.4 − 38.8i)10-s + 1.69e3·11-s + (−31.6 + 18.2i)12-s + (892. − 1.54e3i)13-s + (201. − 348. i)14-s + (−5.43 + 9.40i)15-s + 2.80e3·16-s + (1.21e3 − 2.09e3i)17-s + ⋯
L(s)  = 1  − 0.330i·2-s + (−0.0205 + 0.0118i)3-s + 0.890·4-s + (0.117 − 0.0678i)5-s + (0.00392 + 0.00678i)6-s + (0.384 + 0.221i)7-s − 0.625i·8-s + (−0.499 + 0.865i)9-s + (−0.0224 − 0.0388i)10-s + 1.27·11-s + (−0.0182 + 0.0105i)12-s + (0.406 − 0.703i)13-s + (0.0733 − 0.127i)14-s + (−0.00160 + 0.00278i)15-s + 0.684·16-s + (0.246 − 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.915 + 0.402i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.915 + 0.402i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(2.25585 - 0.474136i\)
\(L(\frac12)\)  \(\approx\)  \(2.25585 - 0.474136i\)
\(L(4)\)   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 \( 1 + (3.81e4 - 6.97e4i)T \)
good2 \( 1 + 2.64iT - 64T^{2} \)
3 \( 1 + (0.554 - 0.320i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (-14.6 + 8.48i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-131. - 76.0i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 - 1.69e3T + 1.77e6T^{2} \)
13 \( 1 + (-892. + 1.54e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (-1.21e3 + 2.09e3i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-8.14e3 + 4.70e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (2.99e3 + 5.19e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-3.63e3 - 2.09e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-1.17e4 - 2.03e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (4.41e4 - 2.54e4i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 6.83e4T + 4.75e9T^{2} \)
47 \( 1 + 1.43e5T + 1.07e10T^{2} \)
53 \( 1 + (-4.02e3 - 6.97e3i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 - 2.70e5T + 4.21e10T^{2} \)
61 \( 1 + (-1.95e5 - 1.12e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (6.97e4 + 1.20e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (5.01e5 + 2.89e5i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (-3.57e5 - 2.06e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (3.87e5 - 6.71e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (2.34e5 + 4.06e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (5.11e5 - 2.95e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.06e6T + 8.32e11T^{2} \)
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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

−14.62514650670263630985114333812, −13.43641192907225825191683418404, −11.87907145363092629225022457686, −11.26257719027802954990085649842, −9.914978389099296433183547103764, −8.314842349064521335129909880910, −6.87070034916201730553972769786, −5.33268285467617490617243987389, −3.13981257756272704237461067606, −1.45075063364234644928626121968, 1.52694279763287481352335941273, 3.64140857416301397120710975872, 5.88460677969532109154710596744, 6.88785994062937501569428136572, 8.387282676244572120792319357239, 9.868908810649277403167269649729, 11.51734646862143666778225962568, 11.97467663937300106918125743358, 14.02564614309540257587924934016, 14.69784988665128775779879501109

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