Properties

Degree 2
Conductor 43
Sign $-0.776 - 0.630i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.75i·2-s + 12.3i·3-s + 8.40·4-s + 21.9i·5-s − 33.9·6-s − 63.3i·7-s + 67.2i·8-s − 71.0·9-s − 60.5·10-s − 1.17·11-s + 103. i·12-s − 173.·13-s + 174.·14-s − 271.·15-s − 51.0·16-s + 469.·17-s + ⋯
L(s)  = 1  + 0.689i·2-s + 1.37i·3-s + 0.525·4-s + 0.879i·5-s − 0.944·6-s − 1.29i·7-s + 1.05i·8-s − 0.877·9-s − 0.605·10-s − 0.00967·11-s + 0.719i·12-s − 1.02·13-s + 0.890·14-s − 1.20·15-s − 0.199·16-s + 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.776 - 0.630i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ -0.776 - 0.630i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.557954 + 1.57290i\)
\(L(\frac12)\)  \(\approx\)  \(0.557954 + 1.57290i\)
\(L(3)\)   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 \( 1 + (1.43e3 + 1.16e3i)T \)
good2 \( 1 - 2.75iT - 16T^{2} \)
3 \( 1 - 12.3iT - 81T^{2} \)
5 \( 1 - 21.9iT - 625T^{2} \)
7 \( 1 + 63.3iT - 2.40e3T^{2} \)
11 \( 1 + 1.17T + 1.46e4T^{2} \)
13 \( 1 + 173.T + 2.85e4T^{2} \)
17 \( 1 - 469.T + 8.35e4T^{2} \)
19 \( 1 + 27.8iT - 1.30e5T^{2} \)
23 \( 1 - 79.3T + 2.79e5T^{2} \)
29 \( 1 + 696. iT - 7.07e5T^{2} \)
31 \( 1 - 1.19e3T + 9.23e5T^{2} \)
37 \( 1 - 1.53e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.45e3T + 2.82e6T^{2} \)
47 \( 1 - 1.69e3T + 4.87e6T^{2} \)
53 \( 1 + 1.99e3T + 7.89e6T^{2} \)
59 \( 1 - 3.56e3T + 1.21e7T^{2} \)
61 \( 1 + 5.44e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.93e3T + 2.01e7T^{2} \)
71 \( 1 + 9.66e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.60e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.14e4T + 3.89e7T^{2} \)
83 \( 1 - 5.52e3T + 4.74e7T^{2} \)
89 \( 1 + 2.55e3iT - 6.27e7T^{2} \)
97 \( 1 - 3.65e3T + 8.85e7T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.55341630590062958172333336082, −14.78979826804969102890607750265, −13.98904015320510274380246628740, −11.74919971998207822593983085432, −10.47636010720746538350222720022, −9.994771621373234759823917643754, −7.84037387907704472559513336440, −6.73123257422785286314077517651, −4.96499534700002864458523298426, −3.26704972758665571293883281250, 1.22828245147751661481197054843, 2.62780511990073890195166163342, 5.51993191471330613894584971890, 7.04123679637500052230556178248, 8.367753879461688962762470226169, 9.880935316537476201243816783775, 11.82387953545417243493547449506, 12.27838502056889228197313882025, 12.94442810054998090173961859859, 14.60894644641096190324918832258

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