Properties

Degree 2
Conductor 43
Sign $0.118 + 0.992i$
Motivic weight 6
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−5.20 + 10.8i)2-s + (3.59 + 0.269i)3-s + (−49.8 − 62.5i)4-s + (−53.9 + 174. i)5-s + (−21.6 + 37.4i)6-s + (85.2 − 49.2i)7-s + (187. − 42.8i)8-s + (−708. − 106. i)9-s + (−1.60e3 − 1.49e3i)10-s + (−88.2 + 110. i)11-s + (−162. − 238. i)12-s + (46.5 − 43.2i)13-s + (88.3 + 1.17e3i)14-s + (−240. + 613. i)15-s + (625. − 2.74e3i)16-s + (4.92e3 − 1.52e3i)17-s + ⋯
L(s)  = 1  + (−0.650 + 1.35i)2-s + (0.133 + 0.00997i)3-s + (−0.779 − 0.977i)4-s + (−0.431 + 1.39i)5-s + (−0.100 + 0.173i)6-s + (0.248 − 0.143i)7-s + (0.366 − 0.0836i)8-s + (−0.971 − 0.146i)9-s + (−1.60 − 1.49i)10-s + (−0.0662 + 0.0830i)11-s + (−0.0940 − 0.137i)12-s + (0.0212 − 0.0196i)13-s + (0.0321 + 0.429i)14-s + (−0.0713 + 0.181i)15-s + (0.152 − 0.669i)16-s + (1.00 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.118 + 0.992i$
motivic weight  =  \(6\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3),\ 0.118 + 0.992i)\)
\(L(\frac{7}{2})\)  \(\approx\)  \(0.209983 - 0.186457i\)
\(L(\frac12)\)  \(\approx\)  \(0.209983 - 0.186457i\)
\(L(4)\)   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 \( 1 + (7.36e4 - 2.98e4i)T \)
good2 \( 1 + (5.20 - 10.8i)T + (-39.9 - 50.0i)T^{2} \)
3 \( 1 + (-3.59 - 0.269i)T + (720. + 108. i)T^{2} \)
5 \( 1 + (53.9 - 174. i)T + (-1.29e4 - 8.80e3i)T^{2} \)
7 \( 1 + (-85.2 + 49.2i)T + (5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (88.2 - 110. i)T + (-3.94e5 - 1.72e6i)T^{2} \)
13 \( 1 + (-46.5 + 43.2i)T + (3.60e5 - 4.81e6i)T^{2} \)
17 \( 1 + (-4.92e3 + 1.52e3i)T + (1.99e7 - 1.35e7i)T^{2} \)
19 \( 1 + (571. + 3.78e3i)T + (-4.49e7 + 1.38e7i)T^{2} \)
23 \( 1 + (-629. - 1.60e3i)T + (-1.08e8 + 1.00e8i)T^{2} \)
29 \( 1 + (2.40e4 - 1.80e3i)T + (5.88e8 - 8.86e7i)T^{2} \)
31 \( 1 + (1.11e4 - 7.57e3i)T + (3.24e8 - 8.26e8i)T^{2} \)
37 \( 1 + (6.78e4 + 3.91e4i)T + (1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + (5.97e4 + 2.87e4i)T + (2.96e9 + 3.71e9i)T^{2} \)
47 \( 1 + (-7.27e4 - 9.11e4i)T + (-2.39e9 + 1.05e10i)T^{2} \)
53 \( 1 + (-8.78e4 - 8.15e4i)T + (1.65e9 + 2.21e10i)T^{2} \)
59 \( 1 + (6.51e4 - 2.85e5i)T + (-3.80e10 - 1.83e10i)T^{2} \)
61 \( 1 + (-1.05e5 + 1.54e5i)T + (-1.88e10 - 4.79e10i)T^{2} \)
67 \( 1 + (9.87e4 - 1.48e4i)T + (8.64e10 - 2.66e10i)T^{2} \)
71 \( 1 + (4.70e5 + 1.84e5i)T + (9.39e10 + 8.71e10i)T^{2} \)
73 \( 1 + (2.43e5 + 2.62e5i)T + (-1.13e10 + 1.50e11i)T^{2} \)
79 \( 1 + (-3.83e5 - 6.63e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-1.75e4 + 2.34e5i)T + (-3.23e11 - 4.87e10i)T^{2} \)
89 \( 1 + (4.25e5 + 3.19e4i)T + (4.91e11 + 7.40e10i)T^{2} \)
97 \( 1 + (1.04e5 - 1.31e5i)T + (-1.85e11 - 8.12e11i)T^{2} \)
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

−15.46644749113382739058283401967, −14.70188196992136218788655883944, −14.01297083357496156819154983275, −11.80549630111299460476796436990, −10.60357804875374140005282179632, −9.099685484690475052053887134987, −7.79212742764380279303873572529, −6.95489158025440967277076862491, −5.62731361577570917845657699410, −3.13313257150685819938226166850, 0.15995203048231139584666233221, 1.65994220642251470876765802682, 3.55084577318836630475000125302, 5.38158333518705497025704795975, 8.194342522566159386068800891969, 8.817437231399615571239626875053, 10.12870045096653268689503212083, 11.55175623350278303513951130140, 12.19528932206408099788900309624, 13.30357613455636145041549243400

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