Properties

Degree 2
Conductor 43
Sign $0.363 + 0.931i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (2.79 + 5.80i)2-s + (−16.1 + 33.5i)3-s + (14.0 − 17.5i)4-s + (−177. − 40.5i)5-s − 239.·6-s − 522. i·7-s + (543. + 123. i)8-s + (−407. − 511. i)9-s + (−261. − 1.14e3i)10-s + (−95.4 − 119. i)11-s + (362. + 752. i)12-s + (−664. + 2.90e3i)13-s + (3.03e3 − 1.46e3i)14-s + (4.22e3 − 5.30e3i)15-s + (479. + 2.10e3i)16-s + (−1.68e3 − 7.36e3i)17-s + ⋯
L(s)  = 1  + (0.349 + 0.725i)2-s + (−0.597 + 1.24i)3-s + (0.218 − 0.274i)4-s + (−1.42 − 0.324i)5-s − 1.10·6-s − 1.52i·7-s + (1.06 + 0.242i)8-s + (−0.559 − 0.701i)9-s + (−0.261 − 1.14i)10-s + (−0.0717 − 0.0899i)11-s + (0.209 + 0.435i)12-s + (−0.302 + 1.32i)13-s + (1.10 − 0.532i)14-s + (1.25 − 1.57i)15-s + (0.117 + 0.512i)16-s + (−0.342 − 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.363 + 0.931i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.363 + 0.931i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.473742 - 0.323709i\)
\(L(\frac12)\)  \(\approx\)  \(0.473742 - 0.323709i\)
\(L(4)\)   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 \( 1 + (-6.22e4 - 4.94e4i)T \)
good2 \( 1 + (-2.79 - 5.80i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (16.1 - 33.5i)T + (-454. - 569. i)T^{2} \)
5 \( 1 + (177. + 40.5i)T + (1.40e4 + 6.77e3i)T^{2} \)
7 \( 1 + 522. iT - 1.17e5T^{2} \)
11 \( 1 + (95.4 + 119. i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (664. - 2.90e3i)T + (-4.34e6 - 2.09e6i)T^{2} \)
17 \( 1 + (1.68e3 + 7.36e3i)T + (-2.17e7 + 1.04e7i)T^{2} \)
19 \( 1 + (2.14e3 + 1.71e3i)T + (1.04e7 + 4.58e7i)T^{2} \)
23 \( 1 + (1.49e4 + 1.87e4i)T + (-3.29e7 + 1.44e8i)T^{2} \)
29 \( 1 + (1.27e4 + 2.64e4i)T + (-3.70e8 + 4.65e8i)T^{2} \)
31 \( 1 + (-5.28e3 + 2.54e3i)T + (5.53e8 - 6.93e8i)T^{2} \)
37 \( 1 - 5.42e4iT - 2.56e9T^{2} \)
41 \( 1 + (1.89e4 - 9.12e3i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (8.81e4 - 1.10e5i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (1.05e4 + 4.63e4i)T + (-1.99e10 + 9.61e9i)T^{2} \)
59 \( 1 + (5.02e4 + 2.20e5i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (8.44e4 - 1.75e5i)T + (-3.21e10 - 4.02e10i)T^{2} \)
67 \( 1 + (2.66e4 - 3.34e4i)T + (-2.01e10 - 8.81e10i)T^{2} \)
71 \( 1 + (-2.33e4 - 1.85e4i)T + (2.85e10 + 1.24e11i)T^{2} \)
73 \( 1 + (2.37e5 + 5.42e4i)T + (1.36e11 + 6.56e10i)T^{2} \)
79 \( 1 + 2.50e5T + 2.43e11T^{2} \)
83 \( 1 + (-5.30e5 - 2.55e5i)T + (2.03e11 + 2.55e11i)T^{2} \)
89 \( 1 + (6.03e4 - 1.25e5i)T + (-3.09e11 - 3.88e11i)T^{2} \)
97 \( 1 + (-2.17e4 - 2.72e4i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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

−14.76794929175985465684443737399, −13.73335667410778097734366265759, −11.66853535540498234858189473299, −10.95728908696985298008612063377, −9.796442369570648502499857854364, −7.80287541338152472058524885393, −6.66169728584511172671350605604, −4.49259128717450403522422946227, −4.37397961183261461226718097149, −0.25152337269475829734258327165, 1.94436938571918976408357462687, 3.54702833386993778169955722450, 5.78276334923538455794997142994, 7.39457156958408793512504668488, 8.216010123323830713355017676166, 10.75203709806953006337910585647, 11.82466716963299974730649418531, 12.31356044536566907858262040897, 13.00250651405636988177165053282, 15.05303782678926919412405898172

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