Properties

Degree 2
Conductor 43
Sign $-0.606 - 0.795i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−27.2 − 34.1i)2-s + (−145. − 21.8i)3-s + (−310. + 1.35e3i)4-s + (−589. − 401. i)5-s + (3.20e3 + 5.54e3i)6-s + (4.83e3 − 8.37e3i)7-s + (3.46e4 − 1.67e4i)8-s + (1.77e3 + 548. i)9-s + (2.32e3 + 3.10e4i)10-s + (−5.25e3 − 2.30e4i)11-s + (7.47e4 − 1.90e5i)12-s + (5.17e3 − 6.90e4i)13-s + (−4.17e5 + 6.29e4i)14-s + (7.67e4 + 7.12e4i)15-s + (−8.71e5 − 4.19e5i)16-s + (4.60e5 − 3.13e5i)17-s + ⋯
L(s)  = 1  + (−1.20 − 1.50i)2-s + (−1.03 − 0.155i)3-s + (−0.605 + 2.65i)4-s + (−0.421 − 0.287i)5-s + (1.00 + 1.74i)6-s + (0.761 − 1.31i)7-s + (2.99 − 1.44i)8-s + (0.0903 + 0.0278i)9-s + (0.0735 + 0.982i)10-s + (−0.108 − 0.473i)11-s + (1.04 − 2.65i)12-s + (0.0502 − 0.670i)13-s + (−2.90 + 0.437i)14-s + (0.391 + 0.363i)15-s + (−3.32 − 1.60i)16-s + (1.33 − 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.606 - 0.795i$
motivic weight  =  \(9\)
character  :  $\chi_{43} (9, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ -0.606 - 0.795i)\)
\(L(5)\)  \(\approx\)  \(0.278908 + 0.563374i\)
\(L(\frac12)\)  \(\approx\)  \(0.278908 + 0.563374i\)
\(L(\frac{11}{2})\)   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 \( 1 + (1.68e7 - 1.47e7i)T \)
good2 \( 1 + (27.2 + 34.1i)T + (-113. + 499. i)T^{2} \)
3 \( 1 + (145. + 21.8i)T + (1.88e4 + 5.80e3i)T^{2} \)
5 \( 1 + (589. + 401. i)T + (7.13e5 + 1.81e6i)T^{2} \)
7 \( 1 + (-4.83e3 + 8.37e3i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (5.25e3 + 2.30e4i)T + (-2.12e9 + 1.02e9i)T^{2} \)
13 \( 1 + (-5.17e3 + 6.90e4i)T + (-1.04e10 - 1.58e9i)T^{2} \)
17 \( 1 + (-4.60e5 + 3.13e5i)T + (4.33e10 - 1.10e11i)T^{2} \)
19 \( 1 + (-5.00e5 + 1.54e5i)T + (2.66e11 - 1.81e11i)T^{2} \)
23 \( 1 + (-1.51e6 + 1.40e6i)T + (1.34e11 - 1.79e12i)T^{2} \)
29 \( 1 + (-1.08e6 + 1.63e5i)T + (1.38e13 - 4.27e12i)T^{2} \)
31 \( 1 + (-7.60e5 + 1.93e6i)T + (-1.93e13 - 1.79e13i)T^{2} \)
37 \( 1 + (1.19e6 + 2.07e6i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 + (-1.25e7 - 1.57e7i)T + (-7.28e13 + 3.19e14i)T^{2} \)
47 \( 1 + (-1.23e7 + 5.42e7i)T + (-1.00e15 - 4.85e14i)T^{2} \)
53 \( 1 + (-3.79e6 - 5.06e7i)T + (-3.26e15 + 4.91e14i)T^{2} \)
59 \( 1 + (-3.20e7 - 1.54e7i)T + (5.40e15 + 6.77e15i)T^{2} \)
61 \( 1 + (5.84e7 + 1.48e8i)T + (-8.57e15 + 7.95e15i)T^{2} \)
67 \( 1 + (-9.23e7 + 2.84e7i)T + (2.24e16 - 1.53e16i)T^{2} \)
71 \( 1 + (-1.48e8 - 1.38e8i)T + (3.42e15 + 4.57e16i)T^{2} \)
73 \( 1 + (-2.06e7 + 2.75e8i)T + (-5.82e16 - 8.77e15i)T^{2} \)
79 \( 1 + (2.03e6 - 3.52e6i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + (2.64e8 + 3.98e7i)T + (1.78e17 + 5.51e16i)T^{2} \)
89 \( 1 + (1.26e8 + 1.90e7i)T + (3.34e17 + 1.03e17i)T^{2} \)
97 \( 1 + (2.88e8 + 1.26e9i)T + (-6.84e17 + 3.29e17i)T^{2} \)
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

−12.54393024156363617940066570019, −11.58839425503071207659698893359, −10.93690332387018024578195623045, −9.984982583333893311052555071218, −8.333761358534073710115971446767, −7.36646985246474583035504785598, −4.77409599439982925438479696797, −3.17123947351558811309663117942, −0.838115936889715251895272166398, −0.65385951416093793182226909270, 1.34191373696171419152980327113, 5.16890373169019901283713513334, 5.78900229670024378006092405387, 7.24428383109592923855537014453, 8.371783914798555835244174036072, 9.576110172936295175964051397729, 10.92537339879109382862629121138, 11.93937246122563628878442451977, 14.31130446701793299499793560280, 15.16223977173752282222005980263

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