Properties

Degree 2
Conductor 43
Sign $0.640 + 0.768i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.815 + 1.02i)2-s + (−202. − 30.5i)3-s + (113. − 497. i)4-s + (626. + 427. i)5-s + (−133. − 232. i)6-s + (−4.73e3 + 8.19e3i)7-s + (1.20e3 − 580. i)8-s + (2.12e4 + 6.55e3i)9-s + (74.1 + 990. i)10-s + (1.33e3 + 5.84e3i)11-s + (−3.81e4 + 9.72e4i)12-s + (−1.71e3 + 2.28e4i)13-s + (−1.22e4 + 1.84e3i)14-s + (−1.13e5 − 1.05e5i)15-s + (−2.33e5 − 1.12e5i)16-s + (−1.68e5 + 1.15e5i)17-s + ⋯
L(s)  = 1  + (0.0360 + 0.0452i)2-s + (−1.44 − 0.217i)3-s + (0.221 − 0.971i)4-s + (0.448 + 0.305i)5-s + (−0.0422 − 0.0730i)6-s + (−0.744 + 1.29i)7-s + (0.104 − 0.0501i)8-s + (1.07 + 0.332i)9-s + (0.00234 + 0.0313i)10-s + (0.0274 + 0.120i)11-s + (−0.531 + 1.35i)12-s + (−0.0166 + 0.221i)13-s + (−0.0852 + 0.0128i)14-s + (−0.580 − 0.538i)15-s + (−0.891 − 0.429i)16-s + (−0.490 + 0.334i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.640 + 0.768i$
motivic weight  =  \(9\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ 0.640 + 0.768i)\)
\(L(5)\)  \(\approx\)  \(0.930599 - 0.435798i\)
\(L(\frac12)\)  \(\approx\)  \(0.930599 - 0.435798i\)
\(L(\frac{11}{2})\)   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 \( 1 + (-1.61e7 + 1.55e7i)T \)
good2 \( 1 + (-0.815 - 1.02i)T + (-113. + 499. i)T^{2} \)
3 \( 1 + (202. + 30.5i)T + (1.88e4 + 5.80e3i)T^{2} \)
5 \( 1 + (-626. - 427. i)T + (7.13e5 + 1.81e6i)T^{2} \)
7 \( 1 + (4.73e3 - 8.19e3i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (-1.33e3 - 5.84e3i)T + (-2.12e9 + 1.02e9i)T^{2} \)
13 \( 1 + (1.71e3 - 2.28e4i)T + (-1.04e10 - 1.58e9i)T^{2} \)
17 \( 1 + (1.68e5 - 1.15e5i)T + (4.33e10 - 1.10e11i)T^{2} \)
19 \( 1 + (-7.47e5 + 2.30e5i)T + (2.66e11 - 1.81e11i)T^{2} \)
23 \( 1 + (-7.29e5 + 6.77e5i)T + (1.34e11 - 1.79e12i)T^{2} \)
29 \( 1 + (-7.54e5 + 1.13e5i)T + (1.38e13 - 4.27e12i)T^{2} \)
31 \( 1 + (-1.84e6 + 4.71e6i)T + (-1.93e13 - 1.79e13i)T^{2} \)
37 \( 1 + (-4.00e6 - 6.92e6i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 + (1.38e7 + 1.74e7i)T + (-7.28e13 + 3.19e14i)T^{2} \)
47 \( 1 + (-7.15e6 + 3.13e7i)T + (-1.00e15 - 4.85e14i)T^{2} \)
53 \( 1 + (-5.21e6 - 6.96e7i)T + (-3.26e15 + 4.91e14i)T^{2} \)
59 \( 1 + (3.53e7 + 1.70e7i)T + (5.40e15 + 6.77e15i)T^{2} \)
61 \( 1 + (2.80e7 + 7.13e7i)T + (-8.57e15 + 7.95e15i)T^{2} \)
67 \( 1 + (-1.84e8 + 5.70e7i)T + (2.24e16 - 1.53e16i)T^{2} \)
71 \( 1 + (-2.48e8 - 2.30e8i)T + (3.42e15 + 4.57e16i)T^{2} \)
73 \( 1 + (6.76e6 - 9.03e7i)T + (-5.82e16 - 8.77e15i)T^{2} \)
79 \( 1 + (-5.17e7 + 8.95e7i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + (-7.18e8 - 1.08e8i)T + (1.78e17 + 5.51e16i)T^{2} \)
89 \( 1 + (-1.11e9 - 1.67e8i)T + (3.34e17 + 1.03e17i)T^{2} \)
97 \( 1 + (7.62e7 + 3.34e8i)T + (-6.84e17 + 3.29e17i)T^{2} \)
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

−13.75185892385688606274222840449, −12.33971962466701465091117112199, −11.45804383659215389032674797657, −10.32677312589337741807701944716, −9.251789518457072754987797763922, −6.72981465295633434828327867367, −6.01384796914579524661249151810, −5.11646944186261824899897344058, −2.31257600700999769586871980733, −0.59147325561247099715603885271, 0.895272466667693519531635727290, 3.43664130279592004291033171732, 4.92392147273485480794153935461, 6.42798730077111367173663907664, 7.50758972178758548311661475113, 9.526781881583286853723744712508, 10.74659747999788334285351630647, 11.70724994639139539079187242103, 12.84941266122340277180221135917, 13.69863749761280940250335013530

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