Properties

Degree 2
Conductor 43
Sign $0.648 - 0.761i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 21.4i·2-s + 198. i·3-s + 563.·4-s + 4.59e3i·5-s + 4.26e3·6-s − 9.95e3i·7-s − 3.40e4i·8-s + 1.95e4·9-s + 9.85e4·10-s + 2.81e5·11-s + 1.11e5i·12-s − 3.87e5·13-s − 2.13e5·14-s − 9.13e5·15-s − 1.54e5·16-s + 7.90e5·17-s + ⋯
L(s)  = 1  − 0.670i·2-s + 0.818i·3-s + 0.549·4-s + 1.46i·5-s + 0.548·6-s − 0.592i·7-s − 1.03i·8-s + 0.330·9-s + 0.985·10-s + 1.74·11-s + 0.450i·12-s − 1.04·13-s − 0.397·14-s − 1.20·15-s − 0.147·16-s + 0.557·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.648 - 0.761i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ 0.648 - 0.761i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(2.36307 + 1.09188i\)
\(L(\frac12)\)  \(\approx\)  \(2.36307 + 1.09188i\)
\(L(6)\)   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 \( 1 + (9.52e7 - 1.11e8i)T \)
good2 \( 1 + 21.4iT - 1.02e3T^{2} \)
3 \( 1 - 198. iT - 5.90e4T^{2} \)
5 \( 1 - 4.59e3iT - 9.76e6T^{2} \)
7 \( 1 + 9.95e3iT - 2.82e8T^{2} \)
11 \( 1 - 2.81e5T + 2.59e10T^{2} \)
13 \( 1 + 3.87e5T + 1.37e11T^{2} \)
17 \( 1 - 7.90e5T + 2.01e12T^{2} \)
19 \( 1 - 2.97e6iT - 6.13e12T^{2} \)
23 \( 1 + 3.73e6T + 4.14e13T^{2} \)
29 \( 1 - 2.26e7iT - 4.20e14T^{2} \)
31 \( 1 - 2.79e7T + 8.19e14T^{2} \)
37 \( 1 - 7.96e6iT - 4.80e15T^{2} \)
41 \( 1 - 5.43e7T + 1.34e16T^{2} \)
47 \( 1 + 6.95e7T + 5.25e16T^{2} \)
53 \( 1 + 3.43e8T + 1.74e17T^{2} \)
59 \( 1 - 4.88e8T + 5.11e17T^{2} \)
61 \( 1 + 8.46e8iT - 7.13e17T^{2} \)
67 \( 1 + 8.00e8T + 1.82e18T^{2} \)
71 \( 1 - 1.08e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.82e9iT - 4.29e18T^{2} \)
79 \( 1 + 4.40e9T + 9.46e18T^{2} \)
83 \( 1 - 4.62e9T + 1.55e19T^{2} \)
89 \( 1 + 7.45e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.42e9T + 7.37e19T^{2} \)
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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

−14.32584616710452615444144498998, −12.31327430915261420454521545677, −11.32677796139889152942549208412, −10.24226281850497931082306282538, −9.792248194508637232723781607688, −7.33681642671507482998209493581, −6.44827257797114816402567404224, −4.06200201214713410382272199053, −3.17786032525239167108412959254, −1.51491409118369235700856384924, 0.910357235244336164511921072597, 2.09853552309808361284545198761, 4.62879539260383400742051394583, 6.05077584663844340841369087977, 7.18359725899822791217778227416, 8.387890205505022759527571389408, 9.523687407603130914737145739481, 11.92619650811302089933370427391, 12.13186767244137904314357014971, 13.56559593618355013551144309765

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