Properties

Label 2-43-43.27-c2-0-1
Degree $2$
Conductor $43$
Sign $-0.413 - 0.910i$
Analytic cond. $1.17166$
Root an. cond. $1.08243$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.53 + 0.807i)2-s + (1.86 + 0.425i)3-s + (8.26 − 3.98i)4-s + (−5.65 + 4.50i)5-s − 6.93·6-s + 11.0i·7-s + (−14.6 + 11.7i)8-s + (−4.82 − 2.32i)9-s + (16.3 − 20.5i)10-s + (8.18 + 3.94i)11-s + (17.0 − 3.89i)12-s + (3.48 + 4.36i)13-s + (−8.95 − 39.2i)14-s + (−12.4 + 5.99i)15-s + (19.6 − 24.6i)16-s + (7.68 − 9.63i)17-s + ⋯
L(s)  = 1  + (−1.76 + 0.403i)2-s + (0.620 + 0.141i)3-s + (2.06 − 0.995i)4-s + (−1.13 + 0.901i)5-s − 1.15·6-s + 1.58i·7-s + (−1.83 + 1.46i)8-s + (−0.535 − 0.257i)9-s + (1.63 − 2.05i)10-s + (0.743 + 0.358i)11-s + (1.42 − 0.324i)12-s + (0.267 + 0.336i)13-s + (−0.639 − 2.80i)14-s + (−0.829 + 0.399i)15-s + (1.22 − 1.53i)16-s + (0.452 − 0.566i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.413 - 0.910i$
Analytic conductor: \(1.17166\)
Root analytic conductor: \(1.08243\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1),\ -0.413 - 0.910i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.258020 + 0.400515i\)
\(L(\frac12)\) \(\approx\) \(0.258020 + 0.400515i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-38.8 - 18.4i)T \)
good2 \( 1 + (3.53 - 0.807i)T + (3.60 - 1.73i)T^{2} \)
3 \( 1 + (-1.86 - 0.425i)T + (8.10 + 3.90i)T^{2} \)
5 \( 1 + (5.65 - 4.50i)T + (5.56 - 24.3i)T^{2} \)
7 \( 1 - 11.0iT - 49T^{2} \)
11 \( 1 + (-8.18 - 3.94i)T + (75.4 + 94.6i)T^{2} \)
13 \( 1 + (-3.48 - 4.36i)T + (-37.6 + 164. i)T^{2} \)
17 \( 1 + (-7.68 + 9.63i)T + (-64.3 - 281. i)T^{2} \)
19 \( 1 + (4.17 + 8.66i)T + (-225. + 282. i)T^{2} \)
23 \( 1 + (-23.3 - 11.2i)T + (329. + 413. i)T^{2} \)
29 \( 1 + (-17.3 + 3.95i)T + (757. - 364. i)T^{2} \)
31 \( 1 + (4.39 + 19.2i)T + (-865. + 416. i)T^{2} \)
37 \( 1 - 16.3iT - 1.36e3T^{2} \)
41 \( 1 + (-10.5 - 46.0i)T + (-1.51e3 + 729. i)T^{2} \)
47 \( 1 + (-12.0 + 5.81i)T + (1.37e3 - 1.72e3i)T^{2} \)
53 \( 1 + (48.3 - 60.5i)T + (-625. - 2.73e3i)T^{2} \)
59 \( 1 + (-33.9 + 42.5i)T + (-774. - 3.39e3i)T^{2} \)
61 \( 1 + (-89.1 - 20.3i)T + (3.35e3 + 1.61e3i)T^{2} \)
67 \( 1 + (53.0 - 25.5i)T + (2.79e3 - 3.50e3i)T^{2} \)
71 \( 1 + (47.5 + 98.7i)T + (-3.14e3 + 3.94e3i)T^{2} \)
73 \( 1 + (-54.6 + 43.6i)T + (1.18e3 - 5.19e3i)T^{2} \)
79 \( 1 + 33.1T + 6.24e3T^{2} \)
83 \( 1 + (-3.02 + 13.2i)T + (-6.20e3 - 2.98e3i)T^{2} \)
89 \( 1 + (-9.70 - 2.21i)T + (7.13e3 + 3.43e3i)T^{2} \)
97 \( 1 + (-29.5 - 14.2i)T + (5.86e3 + 7.35e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.99321683825231870172018293358, −15.18155944728350272219803424055, −14.66737052066026887446249976437, −11.84076839011889191042950462433, −11.24958139590064606281896761979, −9.475338155166899660232609053003, −8.765719022708686461326104296125, −7.65624533225491970320102368870, −6.34744206048874696619742686866, −2.80982802523217977428601274187, 0.886997917757131784754982541999, 3.68159468876314948766241635562, 7.20149545898140203307763171871, 8.176817981574563962431406843244, 8.851782673676090185178165808051, 10.44113620309695698984476264146, 11.35952731252680345880326944786, 12.67666111814262779064824829107, 14.32202063011206748990341484324, 16.00204672013497669182547370813

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