Properties

Degree 2
Conductor 43
Sign $-0.377 - 0.925i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 11.0i·2-s − 50.0i·3-s − 57.2·4-s + 30.3i·5-s − 551.·6-s − 75.1i·7-s − 74.1i·8-s − 1.77e3·9-s + 333.·10-s + 604.·11-s + 2.86e3i·12-s + 2.63e3·13-s − 827.·14-s + 1.51e3·15-s − 4.48e3·16-s + 4.98e3·17-s + ⋯
L(s)  = 1  − 1.37i·2-s − 1.85i·3-s − 0.894·4-s + 0.242i·5-s − 2.55·6-s − 0.219i·7-s − 0.144i·8-s − 2.43·9-s + 0.333·10-s + 0.454·11-s + 1.65i·12-s + 1.19·13-s − 0.301·14-s + 0.449·15-s − 1.09·16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.377 - 0.925i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ -0.377 - 0.925i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.852844 + 1.26927i\)
\(L(\frac12)\)  \(\approx\)  \(0.852844 + 1.26927i\)
\(L(4)\)   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 \( 1 + (-3.00e4 - 7.36e4i)T \)
good2 \( 1 + 11.0iT - 64T^{2} \)
3 \( 1 + 50.0iT - 729T^{2} \)
5 \( 1 - 30.3iT - 1.56e4T^{2} \)
7 \( 1 + 75.1iT - 1.17e5T^{2} \)
11 \( 1 - 604.T + 1.77e6T^{2} \)
13 \( 1 - 2.63e3T + 4.82e6T^{2} \)
17 \( 1 - 4.98e3T + 2.41e7T^{2} \)
19 \( 1 + 3.03e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.24e4T + 1.48e8T^{2} \)
29 \( 1 - 4.44e3iT - 5.94e8T^{2} \)
31 \( 1 + 2.93e4T + 8.87e8T^{2} \)
37 \( 1 + 8.30e4iT - 2.56e9T^{2} \)
41 \( 1 + 3.34e4T + 4.75e9T^{2} \)
47 \( 1 + 1.57e5T + 1.07e10T^{2} \)
53 \( 1 - 2.12e5T + 2.21e10T^{2} \)
59 \( 1 + 2.63e5T + 4.21e10T^{2} \)
61 \( 1 - 1.33e5iT - 5.15e10T^{2} \)
67 \( 1 + 1.57e5T + 9.04e10T^{2} \)
71 \( 1 + 4.85e5iT - 1.28e11T^{2} \)
73 \( 1 - 4.85e4iT - 1.51e11T^{2} \)
79 \( 1 - 7.62e5T + 2.43e11T^{2} \)
83 \( 1 - 2.68e5T + 3.26e11T^{2} \)
89 \( 1 + 1.32e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.43e6T + 8.32e11T^{2} \)
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

−13.41842365017312428017187276904, −12.56397430847526334818805625696, −11.69355018631850048319353974735, −10.72826587238598475614308894133, −8.910242601321135745049043590212, −7.42239510999107766999968725780, −6.17449157945888580298653907289, −3.35705503837447035781849292371, −1.84622879162317828705514349604, −0.75803919748514223617647192878, 3.70050677010692192441423647576, 5.11118008594230860008750925722, 6.12937542079977942229898723948, 8.229856241815648976330009124998, 9.107001507201191564784400154912, 10.39207399791418413615774567024, 11.69593927701318749390995176081, 13.90912453237388092168942836563, 14.79780476998469971375374729811, 15.59124975924227519397348915297

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