Properties

Degree 2
Conductor 43
Sign $-0.687 - 0.726i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.4i·2-s − 477. i·3-s + 914.·4-s − 5.02e3i·5-s − 4.98e3·6-s + 7.30e3i·7-s − 2.02e4i·8-s − 1.68e5·9-s − 5.25e4·10-s − 1.41e5·11-s − 4.36e5i·12-s + 5.66e5·13-s + 7.63e4·14-s − 2.39e6·15-s + 7.25e5·16-s − 3.70e5·17-s + ⋯
L(s)  = 1  − 0.326i·2-s − 1.96i·3-s + 0.893·4-s − 1.60i·5-s − 0.641·6-s + 0.434i·7-s − 0.618i·8-s − 2.86·9-s − 0.525·10-s − 0.879·11-s − 1.75i·12-s + 1.52·13-s + 0.141·14-s − 3.16·15-s + 0.691·16-s − 0.260·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.687 - 0.726i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.687 - 0.726i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.846337 + 1.96728i\)
\(L(\frac12)\)  \(\approx\)  \(0.846337 + 1.96728i\)
\(L(6)\)   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 \( 1 + 10.4iT - 1.02e3T^{2} \)
3 \( 1 + 477. iT - 5.90e4T^{2} \)
5 \( 1 + 5.02e3iT - 9.76e6T^{2} \)
7 \( 1 - 7.30e3iT - 2.82e8T^{2} \)
11 \( 1 + 1.41e5T + 2.59e10T^{2} \)
13 \( 1 - 5.66e5T + 1.37e11T^{2} \)
17 \( 1 + 3.70e5T + 2.01e12T^{2} \)
19 \( 1 + 1.53e6iT - 6.13e12T^{2} \)
23 \( 1 - 9.51e6T + 4.14e13T^{2} \)
29 \( 1 + 1.65e7iT - 4.20e14T^{2} \)
31 \( 1 - 2.42e7T + 8.19e14T^{2} \)
37 \( 1 - 1.62e7iT - 4.80e15T^{2} \)
41 \( 1 + 1.33e8T + 1.34e16T^{2} \)
47 \( 1 + 2.40e8T + 5.25e16T^{2} \)
53 \( 1 + 2.67e7T + 1.74e17T^{2} \)
59 \( 1 - 6.95e8T + 5.11e17T^{2} \)
61 \( 1 - 9.17e8iT - 7.13e17T^{2} \)
67 \( 1 - 1.19e9T + 1.82e18T^{2} \)
71 \( 1 + 4.68e8iT - 3.25e18T^{2} \)
73 \( 1 + 2.00e9iT - 4.29e18T^{2} \)
79 \( 1 + 1.34e9T + 9.46e18T^{2} \)
83 \( 1 + 1.99e9T + 1.55e19T^{2} \)
89 \( 1 + 2.98e8iT - 3.11e19T^{2} \)
97 \( 1 + 1.69e9T + 7.37e19T^{2} \)
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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

−13.07237157566193644020989069411, −12.02609350583114829942981175855, −11.19376086108478826804577521130, −8.794377885163670396790949430471, −7.994590792653838710191693826042, −6.60058083939041294173542403711, −5.47758059280382083885069793298, −2.72142403594638062032229266308, −1.46365608167024137391289069950, −0.69888453517419502166610947746, 2.80662577807578711361791961341, 3.63511196854535479978442920713, 5.48571482195289755437959587009, 6.71834211282800145878868079703, 8.370109871973042794186848249850, 10.14007550097415224341931463456, 10.83339138966256101113879896590, 11.25528027133252914020555848318, 13.93930476017835956106329477954, 14.91111638405226129497112726631

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