Properties

Degree 2
Conductor 43
Sign $0.999 - 0.0406i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 58.7i·2-s + 301. i·3-s − 2.43e3·4-s + 2.78e3i·5-s − 1.77e4·6-s − 4.09e3i·7-s − 8.27e4i·8-s − 3.18e4·9-s − 1.63e5·10-s − 1.23e5·11-s − 7.33e5i·12-s + 3.47e5·13-s + 2.40e5·14-s − 8.38e5·15-s + 2.37e6·16-s − 8.25e4·17-s + ⋯
L(s)  = 1  + 1.83i·2-s + 1.24i·3-s − 2.37·4-s + 0.889i·5-s − 2.27·6-s − 0.243i·7-s − 2.52i·8-s − 0.539·9-s − 1.63·10-s − 0.767·11-s − 2.94i·12-s + 0.935·13-s + 0.447·14-s − 1.10·15-s + 2.26·16-s − 0.0581·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0406i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.999 - 0.0406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.999 - 0.0406i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ 0.999 - 0.0406i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.542145 + 0.0110360i\)
\(L(\frac12)\)  \(\approx\)  \(0.542145 + 0.0110360i\)
\(L(6)\)   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 \( 1 + (1.46e8 - 5.98e6i)T \)
good2 \( 1 - 58.7iT - 1.02e3T^{2} \)
3 \( 1 - 301. iT - 5.90e4T^{2} \)
5 \( 1 - 2.78e3iT - 9.76e6T^{2} \)
7 \( 1 + 4.09e3iT - 2.82e8T^{2} \)
11 \( 1 + 1.23e5T + 2.59e10T^{2} \)
13 \( 1 - 3.47e5T + 1.37e11T^{2} \)
17 \( 1 + 8.25e4T + 2.01e12T^{2} \)
19 \( 1 + 1.73e6iT - 6.13e12T^{2} \)
23 \( 1 + 1.02e7T + 4.14e13T^{2} \)
29 \( 1 - 6.06e6iT - 4.20e14T^{2} \)
31 \( 1 + 4.26e7T + 8.19e14T^{2} \)
37 \( 1 - 1.00e8iT - 4.80e15T^{2} \)
41 \( 1 - 9.51e7T + 1.34e16T^{2} \)
47 \( 1 - 6.18e6T + 5.25e16T^{2} \)
53 \( 1 + 3.72e8T + 1.74e17T^{2} \)
59 \( 1 - 7.30e8T + 5.11e17T^{2} \)
61 \( 1 + 5.55e8iT - 7.13e17T^{2} \)
67 \( 1 + 1.30e9T + 1.82e18T^{2} \)
71 \( 1 + 2.26e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.19e9iT - 4.29e18T^{2} \)
79 \( 1 + 5.10e9T + 9.46e18T^{2} \)
83 \( 1 - 7.51e9T + 1.55e19T^{2} \)
89 \( 1 + 5.82e9iT - 3.11e19T^{2} \)
97 \( 1 + 8.05e9T + 7.37e19T^{2} \)
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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

−15.19448643866463877365790100558, −14.31570554729738175568309429352, −13.26440122067668959905152536609, −10.87167811096986515589922932621, −9.855819969031708819787994987798, −8.599817437265141457315569179398, −7.31064996136421672151056520231, −6.06682666019680176286090132326, −4.80482852234585374804274585974, −3.57736054548185332810493758952, 0.18892030775990726682886710679, 1.32816977273461670570634966865, 2.23691584454939448665750359606, 3.96433305317155919118450808455, 5.64370666633314046110523613076, 7.937340581778786160595337321598, 8.973100946214492308352026820026, 10.38625053087392653865589556616, 11.67038209973287533286545825641, 12.60800767824383367724359691402

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