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Properties

Label 2-43-43.36-c3-0-9
Degree $2$
Conductor $43$
Sign $-0.392 + 0.919i$
Analytic cond. $2.53708$
Root an. cond. $1.59282$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.92·2-s + (−4.18 − 7.25i)3-s − 4.30·4-s + (−0.0351 − 0.0608i)5-s + (−8.05 − 13.9i)6-s + (11.7 − 20.3i)7-s − 23.6·8-s + (−21.5 + 37.4i)9-s + (−0.0675 − 0.116i)10-s + 55.0·11-s + (18.0 + 31.2i)12-s + (13.8 − 24.0i)13-s + (22.5 − 39.0i)14-s + (−0.294 + 0.509i)15-s − 11.0·16-s + (−14.7 + 25.4i)17-s + ⋯
L(s)  = 1  + 0.679·2-s + (−0.806 − 1.39i)3-s − 0.537·4-s + (−0.00314 − 0.00543i)5-s + (−0.548 − 0.949i)6-s + (0.632 − 1.09i)7-s − 1.04·8-s + (−0.799 + 1.38i)9-s + (−0.00213 − 0.00369i)10-s + 1.50·11-s + (0.433 + 0.751i)12-s + (0.296 − 0.513i)13-s + (0.430 − 0.745i)14-s + (−0.00506 + 0.00877i)15-s − 0.172·16-s + (−0.209 + 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.392 + 0.919i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ -0.392 + 0.919i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.672760 - 1.01858i\)
\(L(\frac12)\) \(\approx\) \(0.672760 - 1.01858i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (256. - 116. i)T \)
good2 \( 1 - 1.92T + 8T^{2} \)
3 \( 1 + (4.18 + 7.25i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (0.0351 + 0.0608i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-11.7 + 20.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 - 55.0T + 1.33e3T^{2} \)
13 \( 1 + (-13.8 + 24.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (14.7 - 25.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-22.9 - 39.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (63.3 + 109. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (67.6 - 117. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-109. - 188. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (185. + 320. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 357.T + 6.89e4T^{2} \)
47 \( 1 - 442.T + 1.03e5T^{2} \)
53 \( 1 + (-139. - 241. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 413.T + 2.05e5T^{2} \)
61 \( 1 + (280. - 485. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (89.6 + 155. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (295. - 512. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (352. - 610. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-298. + 517. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (18.9 + 32.8i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-188. - 325. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.65e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.42210849067353328601961226760, −13.92640101818868406003293271549, −12.69617932369784087194176366132, −11.99877614397750111001724248767, −10.65621778126143809035477350411, −8.556129171394203820305714417624, −7.06476717127408202380038940701, −5.89919667998008422529139543496, −4.20106696642944900003713661653, −1.01509918521152322684119863858, 3.86102580054454294646589876334, 4.98477507347379189273083031848, 6.05052063784567570962964643009, 8.923144245675810697788472397642, 9.585618760615846962182305211577, 11.50641634775242156484483624024, 11.84597801092294992811762145993, 13.72279152587652169868519970018, 14.92247972504935904304835919835, 15.49183792272719018233843087007

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