Properties

Degree 2
Conductor 43
Sign $-0.915 - 0.403i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 41.2i·2-s + (−198. + 114. i)3-s − 680.·4-s + (422. − 244. i)5-s + (4.72e3 + 8.18e3i)6-s + (5.42e3 + 3.12e3i)7-s − 1.41e4i·8-s + (−3.30e3 + 5.72e3i)9-s + (−1.00e4 − 1.74e4i)10-s + 1.24e5·11-s + (1.34e5 − 7.78e4i)12-s + (2.55e5 − 4.42e5i)13-s + (1.29e5 − 2.23e5i)14-s + (−5.58e4 + 9.67e4i)15-s − 1.28e6·16-s + (5.87e5 − 1.01e6i)17-s + ⋯
L(s)  = 1  − 1.29i·2-s + (−0.816 + 0.471i)3-s − 0.664·4-s + (0.135 − 0.0780i)5-s + (0.607 + 1.05i)6-s + (0.322 + 0.186i)7-s − 0.433i·8-s + (−0.0559 + 0.0969i)9-s + (−0.100 − 0.174i)10-s + 0.770·11-s + (0.542 − 0.312i)12-s + (0.687 − 1.19i)13-s + (0.240 − 0.416i)14-s + (−0.0735 + 0.127i)15-s − 1.22·16-s + (0.413 − 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.915 - 0.403i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.915 - 0.403i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.160846 + 0.764116i\)
\(L(\frac12)\)  \(\approx\)  \(0.160846 + 0.764116i\)
\(L(6)\)   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 \( 1 + (-7.06e7 + 1.28e8i)T \)
good2 \( 1 + 41.2iT - 1.02e3T^{2} \)
3 \( 1 + (198. - 114. i)T + (2.95e4 - 5.11e4i)T^{2} \)
5 \( 1 + (-422. + 244. i)T + (4.88e6 - 8.45e6i)T^{2} \)
7 \( 1 + (-5.42e3 - 3.12e3i)T + (1.41e8 + 2.44e8i)T^{2} \)
11 \( 1 - 1.24e5T + 2.59e10T^{2} \)
13 \( 1 + (-2.55e5 + 4.42e5i)T + (-6.89e10 - 1.19e11i)T^{2} \)
17 \( 1 + (-5.87e5 + 1.01e6i)T + (-1.00e12 - 1.74e12i)T^{2} \)
19 \( 1 + (2.76e6 - 1.59e6i)T + (3.06e12 - 5.30e12i)T^{2} \)
23 \( 1 + (4.59e6 + 7.96e6i)T + (-2.07e13 + 3.58e13i)T^{2} \)
29 \( 1 + (5.50e6 + 3.17e6i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (-2.72e7 - 4.71e7i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 + (2.51e7 - 1.45e7i)T + (2.40e15 - 4.16e15i)T^{2} \)
41 \( 1 + 1.18e8T + 1.34e16T^{2} \)
47 \( 1 + 1.76e8T + 5.25e16T^{2} \)
53 \( 1 + (3.34e8 + 5.79e8i)T + (-8.74e16 + 1.51e17i)T^{2} \)
59 \( 1 + 9.34e8T + 5.11e17T^{2} \)
61 \( 1 + (7.82e8 + 4.51e8i)T + (3.56e17 + 6.17e17i)T^{2} \)
67 \( 1 + (6.25e8 + 1.08e9i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 + (-5.58e8 - 3.22e8i)T + (1.62e18 + 2.81e18i)T^{2} \)
73 \( 1 + (-1.34e9 - 7.74e8i)T + (2.14e18 + 3.72e18i)T^{2} \)
79 \( 1 + (3.38e8 - 5.87e8i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (1.70e9 + 2.95e9i)T + (-7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 + (-3.99e9 + 2.30e9i)T + (1.55e19 - 2.70e19i)T^{2} \)
97 \( 1 + 4.35e9T + 7.37e19T^{2} \)
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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

−12.58682158768552974628698480222, −11.72281138336442395582511139294, −10.74371549580913476916711693299, −10.02740876607833450855785563810, −8.432256317895312908437081153156, −6.29390033423485941145342970301, −4.85166493061846842076504611768, −3.39295666415239507128357572830, −1.73141797136489226784625970295, −0.27918073497552652960795056083, 1.60569229103611182827188491894, 4.30878499028865309354616036571, 6.05559816344827740152796527560, 6.45693141288003511171597517776, 7.84908492609820987190911069965, 9.171106448682595766888543852684, 11.11870550294076552440253189560, 11.94095490387595598075681530609, 13.57674250051321682542409362444, 14.53909081913616073505068414485

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