Properties

Degree 2
Conductor 43
Sign $0.975 + 0.221i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.0i·2-s + (−28.7 + 16.5i)3-s − 134.·4-s + (147. − 85.1i)5-s + (−233. − 404. i)6-s + (−245. − 141. i)7-s − 990. i·8-s + (185. − 321. i)9-s + (1.19e3 + 2.07e3i)10-s − 907.·11-s + (3.85e3 − 2.22e3i)12-s + (1.02e3 − 1.77e3i)13-s + (1.99e3 − 3.45e3i)14-s + (−2.82e3 + 4.89e3i)15-s + 5.35e3·16-s + (−3.80e3 + 6.58e3i)17-s + ⋯
L(s)  = 1  + 1.76i·2-s + (−1.06 + 0.614i)3-s − 2.09·4-s + (1.17 − 0.681i)5-s + (−1.08 − 1.87i)6-s + (−0.714 − 0.412i)7-s − 1.93i·8-s + (0.254 − 0.440i)9-s + (1.19 + 2.07i)10-s − 0.682·11-s + (2.23 − 1.28i)12-s + (0.465 − 0.806i)13-s + (0.726 − 1.25i)14-s + (−0.836 + 1.44i)15-s + 1.30·16-s + (−0.774 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.975 + 0.221i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.975 + 0.221i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.332955 - 0.0373457i\)
\(L(\frac12)\)  \(\approx\)  \(0.332955 - 0.0373457i\)
\(L(4)\)   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 \( 1 + (2.42e4 - 7.57e4i)T \)
good2 \( 1 - 14.0iT - 64T^{2} \)
3 \( 1 + (28.7 - 16.5i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (-147. + 85.1i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (245. + 141. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + 907.T + 1.77e6T^{2} \)
13 \( 1 + (-1.02e3 + 1.77e3i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 + (3.80e3 - 6.58e3i)T + (-1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-5.71e3 + 3.29e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (8.93e3 + 1.54e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-1.98e4 - 1.14e4i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (2.38e4 + 4.12e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-4.44e4 + 2.56e4i)T + (1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 5.71e4T + 4.75e9T^{2} \)
47 \( 1 + 1.46e5T + 1.07e10T^{2} \)
53 \( 1 + (-5.71e4 - 9.90e4i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + 9.38e4T + 4.21e10T^{2} \)
61 \( 1 + (1.71e5 + 9.92e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-8.24e4 - 1.42e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + (5.93e5 + 3.42e5i)T + (6.40e10 + 1.10e11i)T^{2} \)
73 \( 1 + (9.40e3 + 5.42e3i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-2.01e5 + 3.48e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-1.34e5 - 2.32e5i)T + (-1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + (-3.41e5 + 1.97e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.75e6T + 8.32e11T^{2} \)
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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

−15.06610283810350279186094422214, −13.52312045821700015581571536080, −12.92532320649966775036150895590, −10.59326780593798890654981119180, −9.593379966511662466659282081018, −8.210365662938662904427689085155, −6.35090631312852723114035549176, −5.71464054408285876858552748239, −4.60794277345096485017166675151, −0.18033646965567431426527912959, 1.61527687417267778772885296975, 2.99533667357648405920264161640, 5.35026623126043481343703448005, 6.64947740024796144682703760969, 9.302765898306571656732171189582, 10.15084597228134167656218632533, 11.33493248610619376518234884252, 12.06056531484887173713238697330, 13.30225303650434177060306646353, 13.88011403339434994519780257123

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