Properties

Degree 2
Conductor 43
Sign $0.662 - 0.748i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 50.9i·2-s + (10.9 − 6.31i)3-s − 1.57e3·4-s + (3.08e3 − 1.77e3i)5-s + (−322. − 557. i)6-s + (5.34e3 + 3.08e3i)7-s + 2.80e4i·8-s + (−2.94e4 + 5.09e4i)9-s + (−9.06e4 − 1.57e5i)10-s − 1.87e5·11-s + (−1.72e4 + 9.94e3i)12-s + (−1.15e5 + 1.99e5i)13-s + (1.57e5 − 2.72e5i)14-s + (2.24e4 − 3.89e4i)15-s − 1.82e5·16-s + (−6.68e5 + 1.15e6i)17-s + ⋯
L(s)  = 1  − 1.59i·2-s + (0.0450 − 0.0260i)3-s − 1.53·4-s + (0.986 − 0.569i)5-s + (−0.0414 − 0.0717i)6-s + (0.318 + 0.183i)7-s + 0.855i·8-s + (−0.498 + 0.863i)9-s + (−0.906 − 1.57i)10-s − 1.16·11-s + (−0.0692 + 0.0399i)12-s + (−0.310 + 0.537i)13-s + (0.292 − 0.506i)14-s + (0.0296 − 0.0512i)15-s − 0.173·16-s + (−0.470 + 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.748i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.662 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.662 - 0.748i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ 0.662 - 0.748i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.236391 + 0.106428i\)
\(L(\frac12)\)  \(\approx\)  \(0.236391 + 0.106428i\)
\(L(6)\)   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 \( 1 + (-1.01e8 - 1.06e8i)T \)
good2 \( 1 + 50.9iT - 1.02e3T^{2} \)
3 \( 1 + (-10.9 + 6.31i)T + (2.95e4 - 5.11e4i)T^{2} \)
5 \( 1 + (-3.08e3 + 1.77e3i)T + (4.88e6 - 8.45e6i)T^{2} \)
7 \( 1 + (-5.34e3 - 3.08e3i)T + (1.41e8 + 2.44e8i)T^{2} \)
11 \( 1 + 1.87e5T + 2.59e10T^{2} \)
13 \( 1 + (1.15e5 - 1.99e5i)T + (-6.89e10 - 1.19e11i)T^{2} \)
17 \( 1 + (6.68e5 - 1.15e6i)T + (-1.00e12 - 1.74e12i)T^{2} \)
19 \( 1 + (2.62e6 - 1.51e6i)T + (3.06e12 - 5.30e12i)T^{2} \)
23 \( 1 + (7.08e4 + 1.22e5i)T + (-2.07e13 + 3.58e13i)T^{2} \)
29 \( 1 + (5.31e6 + 3.06e6i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (1.91e7 + 3.31e7i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 + (-1.03e8 + 5.98e7i)T + (2.40e15 - 4.16e15i)T^{2} \)
41 \( 1 + 2.68e7T + 1.34e16T^{2} \)
47 \( 1 + 2.46e8T + 5.25e16T^{2} \)
53 \( 1 + (-5.63e7 - 9.75e7i)T + (-8.74e16 + 1.51e17i)T^{2} \)
59 \( 1 + 2.95e8T + 5.11e17T^{2} \)
61 \( 1 + (-1.33e8 - 7.68e7i)T + (3.56e17 + 6.17e17i)T^{2} \)
67 \( 1 + (-9.63e8 - 1.66e9i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 + (9.64e8 + 5.56e8i)T + (1.62e18 + 2.81e18i)T^{2} \)
73 \( 1 + (3.47e8 + 2.00e8i)T + (2.14e18 + 3.72e18i)T^{2} \)
79 \( 1 + (-2.98e8 + 5.16e8i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (2.97e9 + 5.14e9i)T + (-7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 + (5.88e9 - 3.39e9i)T + (1.55e19 - 2.70e19i)T^{2} \)
97 \( 1 - 8.08e9T + 7.37e19T^{2} \)
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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

−13.23931891441126749896685116788, −12.90090790347897146988617130606, −11.35459337860614430954034431451, −10.48170171997108660172440335257, −9.401635012402846441684680858980, −8.174244514786956232606396832963, −5.69707867764489947260239815866, −4.40336105668502088307880794303, −2.41441092011414949218073526743, −1.79010178927852981251013901603, 0.07529207517973025118418784819, 2.62808588720358663899989713449, 4.92623591821688202492653066933, 6.03133297263790932443714740904, 7.03597792606990654515517002362, 8.343151285139521195215547123952, 9.575730258414704309936943361614, 10.96802184740263202828962484457, 12.97515708880924336328562146403, 14.01539167501882479380342445192

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