Properties

Degree 2
Conductor 43
Sign $-0.836 + 0.548i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.72i·2-s + 8.31i·3-s − 29.1·4-s − 1.48i·5-s − 55.9·6-s + 13.7i·7-s − 88.6i·8-s + 11.7·9-s + 9.96·10-s − 10.2·11-s − 242. i·12-s + 98.4·13-s − 92.7·14-s + 12.3·15-s + 128.·16-s − 286.·17-s + ⋯
L(s)  = 1  + 1.68i·2-s + 0.924i·3-s − 1.82·4-s − 0.0592i·5-s − 1.55·6-s + 0.281i·7-s − 1.38i·8-s + 0.145·9-s + 0.0996·10-s − 0.0843·11-s − 1.68i·12-s + 0.582·13-s − 0.473·14-s + 0.0548·15-s + 0.503·16-s − 0.991·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.836 + 0.548i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ -0.836 + 0.548i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.357835 - 1.19851i\)
\(L(\frac12)\)  \(\approx\)  \(0.357835 - 1.19851i\)
\(L(3)\)   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 \( 1 + (1.54e3 - 1.01e3i)T \)
good2 \( 1 - 6.72iT - 16T^{2} \)
3 \( 1 - 8.31iT - 81T^{2} \)
5 \( 1 + 1.48iT - 625T^{2} \)
7 \( 1 - 13.7iT - 2.40e3T^{2} \)
11 \( 1 + 10.2T + 1.46e4T^{2} \)
13 \( 1 - 98.4T + 2.85e4T^{2} \)
17 \( 1 + 286.T + 8.35e4T^{2} \)
19 \( 1 - 367. iT - 1.30e5T^{2} \)
23 \( 1 + 242.T + 2.79e5T^{2} \)
29 \( 1 - 1.14e3iT - 7.07e5T^{2} \)
31 \( 1 - 895.T + 9.23e5T^{2} \)
37 \( 1 + 2.29e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.69e3T + 2.82e6T^{2} \)
47 \( 1 - 743.T + 4.87e6T^{2} \)
53 \( 1 - 99.3T + 7.89e6T^{2} \)
59 \( 1 - 3.28e3T + 1.21e7T^{2} \)
61 \( 1 + 3.22e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.55e3T + 2.01e7T^{2} \)
71 \( 1 - 2.95e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.61e3iT - 2.83e7T^{2} \)
79 \( 1 - 2.38e3T + 3.89e7T^{2} \)
83 \( 1 + 6.42e3T + 4.74e7T^{2} \)
89 \( 1 - 3.29e3iT - 6.27e7T^{2} \)
97 \( 1 + 9.32e3T + 8.85e7T^{2} \)
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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

−15.97215117054026921247706329140, −15.05471202278874009471985218741, −14.13597307783280004114507813875, −12.77426851382335095228552720984, −10.78493141643187849872303521215, −9.357884716054689167804915182441, −8.357888874946987861245732131176, −6.83959549429054552194910530726, −5.46627791896403259791043817940, −4.16893775230296432454822732375, 0.878613051298174907659235221261, 2.46503354508459468744653875224, 4.33869015343380834292318253105, 6.74629077503773298087293372423, 8.497208705492321835433251939389, 9.944989932053398330528920435852, 11.10776563197158773319785862571, 12.06748767970320461597982074378, 13.23966356583276260910784400502, 13.63409719513973738329664304696

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