Properties

Degree 2
Conductor 43
Sign $-0.798 + 0.601i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.266i·2-s + (−177. + 102. i)3-s + 1.02e3·4-s + (−1.11e3 + 642. i)5-s + (27.2 + 47.1i)6-s + (−3.82e3 − 2.20e3i)7-s − 544. i·8-s + (−8.61e3 + 1.49e4i)9-s + (170. + 295. i)10-s + 1.42e4·11-s + (−1.81e5 + 1.04e5i)12-s + (3.66e3 − 6.34e3i)13-s + (−587. + 1.01e3i)14-s + (1.31e5 − 2.27e5i)15-s + 1.04e6·16-s + (−9.20e5 + 1.59e6i)17-s + ⋯
L(s)  = 1  − 0.00831i·2-s + (−0.728 + 0.420i)3-s + 0.999·4-s + (−0.355 + 0.205i)5-s + (0.00349 + 0.00606i)6-s + (−0.227 − 0.131i)7-s − 0.0166i·8-s + (−0.145 + 0.252i)9-s + (0.00170 + 0.00295i)10-s + 0.0885·11-s + (−0.728 + 0.420i)12-s + (0.00987 − 0.0170i)13-s + (−0.00109 + 0.00189i)14-s + (0.172 − 0.299i)15-s + 0.999·16-s + (−0.648 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.798 + 0.601i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.798 + 0.601i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.0198034 - 0.0592038i\)
\(L(\frac12)\)  \(\approx\)  \(0.0198034 - 0.0592038i\)
\(L(6)\)   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 \( 1 + (7.80e7 + 1.24e8i)T \)
good2 \( 1 + 0.266iT - 1.02e3T^{2} \)
3 \( 1 + (177. - 102. i)T + (2.95e4 - 5.11e4i)T^{2} \)
5 \( 1 + (1.11e3 - 642. i)T + (4.88e6 - 8.45e6i)T^{2} \)
7 \( 1 + (3.82e3 + 2.20e3i)T + (1.41e8 + 2.44e8i)T^{2} \)
11 \( 1 - 1.42e4T + 2.59e10T^{2} \)
13 \( 1 + (-3.66e3 + 6.34e3i)T + (-6.89e10 - 1.19e11i)T^{2} \)
17 \( 1 + (9.20e5 - 1.59e6i)T + (-1.00e12 - 1.74e12i)T^{2} \)
19 \( 1 + (-1.53e6 + 8.88e5i)T + (3.06e12 - 5.30e12i)T^{2} \)
23 \( 1 + (4.35e6 + 7.53e6i)T + (-2.07e13 + 3.58e13i)T^{2} \)
29 \( 1 + (2.56e7 + 1.48e7i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (1.70e7 + 2.94e7i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 + (5.50e7 - 3.17e7i)T + (2.40e15 - 4.16e15i)T^{2} \)
41 \( 1 - 3.59e7T + 1.34e16T^{2} \)
47 \( 1 + 1.70e8T + 5.25e16T^{2} \)
53 \( 1 + (2.99e8 + 5.17e8i)T + (-8.74e16 + 1.51e17i)T^{2} \)
59 \( 1 + 5.02e8T + 5.11e17T^{2} \)
61 \( 1 + (-1.11e9 - 6.41e8i)T + (3.56e17 + 6.17e17i)T^{2} \)
67 \( 1 + (-8.47e8 - 1.46e9i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 + (-1.93e9 - 1.11e9i)T + (1.62e18 + 2.81e18i)T^{2} \)
73 \( 1 + (1.99e9 + 1.15e9i)T + (2.14e18 + 3.72e18i)T^{2} \)
79 \( 1 + (3.14e8 - 5.45e8i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (-4.84e8 - 8.39e8i)T + (-7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 + (5.24e9 - 3.02e9i)T + (1.55e19 - 2.70e19i)T^{2} \)
97 \( 1 - 9.46e9T + 7.37e19T^{2} \)
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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

−13.10496059154287729124690528998, −11.69576755751828159872195153418, −11.05110189766707120361212405984, −10.01434097764958964032402121259, −8.080928606995119077830140179994, −6.72403250038712615295496299602, −5.57209791480549076358030969968, −3.81987954929593308889968622131, −2.08754952024542213164967828347, −0.02053036098640656422058068041, 1.55011173227598682933433846953, 3.31484832770748859085477563335, 5.43073323774942699460977909613, 6.59336737963617585024921155098, 7.63518760011175698130572632746, 9.410082535081017865185933137941, 11.05248627733947152826761609067, 11.79379916881748837897060896845, 12.63254939601361202048252722112, 14.25584377637059637103372337192

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