Properties

Label 2-43-43.12-c8-0-2
Degree $2$
Conductor $43$
Sign $-0.576 - 0.817i$
Analytic cond. $17.5172$
Root an. cond. $4.18536$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−12.0 − 25.0i)2-s + (83.1 + 121. i)3-s + (−322. + 404. i)4-s + (−634. + 684. i)5-s + (2.05e3 − 3.55e3i)6-s + (2.81e3 − 1.62e3i)7-s + (7.08e3 + 1.61e3i)8-s + (−5.55e3 + 1.41e4i)9-s + (2.48e4 + 7.65e3i)10-s + (−5.93e3 − 7.43e3i)11-s + (−7.61e4 − 5.70e3i)12-s + (2.46e4 − 7.60e3i)13-s + (−7.46e4 − 5.09e4i)14-s + (−1.36e5 − 2.05e4i)15-s + (−1.55e4 − 6.79e4i)16-s + (−7.22e4 + 6.70e4i)17-s + ⋯
L(s)  = 1  + (−0.754 − 1.56i)2-s + (1.02 + 1.50i)3-s + (−1.26 + 1.58i)4-s + (−1.01 + 1.09i)5-s + (1.58 − 2.74i)6-s + (1.17 − 0.676i)7-s + (1.72 + 0.394i)8-s + (−0.847 + 2.15i)9-s + (2.48 + 0.765i)10-s + (−0.405 − 0.507i)11-s + (−3.67 − 0.275i)12-s + (0.863 − 0.266i)13-s + (−1.94 − 1.32i)14-s + (−2.69 − 0.405i)15-s + (−0.236 − 1.03i)16-s + (−0.865 + 0.802i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.576 - 0.817i$
Analytic conductor: \(17.5172\)
Root analytic conductor: \(4.18536\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :4),\ -0.576 - 0.817i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.307202 + 0.592733i\)
\(L(\frac12)\) \(\approx\) \(0.307202 + 0.592733i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (2.39e6 + 2.44e6i)T \)
good2 \( 1 + (12.0 + 25.0i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (-83.1 - 121. i)T + (-2.39e3 + 6.10e3i)T^{2} \)
5 \( 1 + (634. - 684. i)T + (-2.91e4 - 3.89e5i)T^{2} \)
7 \( 1 + (-2.81e3 + 1.62e3i)T + (2.88e6 - 4.99e6i)T^{2} \)
11 \( 1 + (5.93e3 + 7.43e3i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (-2.46e4 + 7.60e3i)T + (6.73e8 - 4.59e8i)T^{2} \)
17 \( 1 + (7.22e4 - 6.70e4i)T + (5.21e8 - 6.95e9i)T^{2} \)
19 \( 1 + (1.85e5 - 7.26e4i)T + (1.24e10 - 1.15e10i)T^{2} \)
23 \( 1 + (2.01e5 - 3.04e4i)T + (7.48e10 - 2.30e10i)T^{2} \)
29 \( 1 + (6.96e4 - 1.02e5i)T + (-1.82e11 - 4.65e11i)T^{2} \)
31 \( 1 + (-7.64e4 + 1.01e6i)T + (-8.43e11 - 1.27e11i)T^{2} \)
37 \( 1 + (8.72e5 + 5.03e5i)T + (1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + (2.52e6 - 1.21e6i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (-1.42e6 + 1.78e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (2.02e6 + 6.25e5i)T + (5.14e13 + 3.50e13i)T^{2} \)
59 \( 1 + (-3.25e6 - 1.42e7i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (-1.20e7 + 9.03e5i)T + (1.89e14 - 2.85e13i)T^{2} \)
67 \( 1 + (-2.45e6 - 6.24e6i)T + (-2.97e14 + 2.76e14i)T^{2} \)
71 \( 1 + (-8.79e5 + 5.83e6i)T + (-6.17e14 - 1.90e14i)T^{2} \)
73 \( 1 + (-7.32e6 - 2.37e7i)T + (-6.66e14 + 4.54e14i)T^{2} \)
79 \( 1 + (-1.48e7 - 2.57e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + (5.56e7 - 3.79e7i)T + (8.22e14 - 2.09e15i)T^{2} \)
89 \( 1 + (-4.94e7 - 7.24e7i)T + (-1.43e15 + 3.66e15i)T^{2} \)
97 \( 1 + (-1.04e7 - 1.30e7i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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

−14.69475953022220334560613644407, −13.48281079071800543554744014639, −11.37880656572206533945417237597, −10.76840877535550272143396337392, −10.28573632340855237509400667676, −8.498221060021292822829998812930, −8.110158580243354757608067991668, −4.11611769079421901076269571190, −3.66004103978298276455207044064, −2.21385277639941785911892727286, 0.28380305576427354288992903087, 1.74904024632555216359409339274, 4.79544785120698086890742975404, 6.60166393276730352051093923993, 7.74258506165871603192370627867, 8.549583181268213443844729023088, 8.802570215921026122589201847569, 11.73471589823265597973021242581, 12.93676899020565845579028707883, 14.10351320810802045731226990328

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