Properties

Degree 2
Conductor 43
Sign $0.998 + 0.0467i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3.31·2-s + (64.4 − 111. i)3-s − 501.·4-s + (−994. + 1.72e3i)5-s + (213. − 369. i)6-s + (−3.83e3 − 6.64e3i)7-s − 3.35e3·8-s + (1.53e3 + 2.66e3i)9-s + (−3.29e3 + 5.70e3i)10-s + 5.68e4·11-s + (−3.22e4 + 5.59e4i)12-s + (3.14e4 + 5.44e4i)13-s + (−1.26e4 − 2.19e4i)14-s + (1.28e5 + 2.21e5i)15-s + 2.45e5·16-s + (−3.32e4 − 5.76e4i)17-s + ⋯
L(s)  = 1  + 0.146·2-s + (0.459 − 0.795i)3-s − 0.978·4-s + (−0.711 + 1.23i)5-s + (0.0672 − 0.116i)6-s + (−0.603 − 1.04i)7-s − 0.289·8-s + (0.0780 + 0.135i)9-s + (−0.104 + 0.180i)10-s + 1.17·11-s + (−0.449 + 0.778i)12-s + (0.305 + 0.529i)13-s + (−0.0883 − 0.152i)14-s + (0.653 + 1.13i)15-s + 0.936·16-s + (−0.0966 − 0.167i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0467i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.998 + 0.0467i$
motivic weight  =  \(9\)
character  :  $\chi_{43} (6, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ 0.998 + 0.0467i)\)
\(L(5)\)  \(\approx\)  \(1.60746 - 0.0375973i\)
\(L(\frac12)\)  \(\approx\)  \(1.60746 - 0.0375973i\)
\(L(\frac{11}{2})\)   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 \( 1 + (1.72e7 - 1.43e7i)T \)
good2 \( 1 - 3.31T + 512T^{2} \)
3 \( 1 + (-64.4 + 111. i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (994. - 1.72e3i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 + (3.83e3 + 6.64e3i)T + (-2.01e7 + 3.49e7i)T^{2} \)
11 \( 1 - 5.68e4T + 2.35e9T^{2} \)
13 \( 1 + (-3.14e4 - 5.44e4i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + (3.32e4 + 5.76e4i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-3.43e5 + 5.95e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (9.37e4 - 1.62e5i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + (-2.36e6 - 4.08e6i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 + (-4.19e5 + 7.27e5i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (1.90e6 - 3.30e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 1.88e7T + 3.27e14T^{2} \)
47 \( 1 - 4.22e7T + 1.11e15T^{2} \)
53 \( 1 + (-1.81e7 + 3.13e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 - 9.07e5T + 8.66e15T^{2} \)
61 \( 1 + (-8.54e7 - 1.48e8i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (6.89e7 - 1.19e8i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + (-1.65e8 - 2.86e8i)T + (-2.29e16 + 3.97e16i)T^{2} \)
73 \( 1 + (2.35e8 + 4.07e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (2.38e8 + 4.12e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 + (1.45e8 - 2.51e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (-2.84e8 + 4.92e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 - 2.47e8T + 7.60e17T^{2} \)
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

−13.93228687042492937200842643641, −13.14854052691557659726075166078, −11.68312725103652351145190093309, −10.33316682662104114905438329952, −8.921372887711111450031656691778, −7.39913734116949345828790601603, −6.70789228283168886219041566341, −4.24155329708584193625682254908, −3.14405413464721396063034405576, −0.925933684438792542281274809255, 0.78667954431665925220264675613, 3.52865768723203613448976831765, 4.38960611368219431029416666254, 5.79607832564924013817606066987, 8.333307943829468697200811856318, 9.032555704079270588567090468844, 9.821283252154559946456003832299, 12.07736095569109160450633814135, 12.60163371210002501046423587269, 14.07940707138921813763533756796

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