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Properties

Label 2-43-43.8-c2-0-3
Degree $2$
Conductor $43$
Sign $0.846 - 0.532i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.62 + 0.826i)2-s + (−3.87 + 0.884i)3-s + (8.83 + 4.25i)4-s + (−2.82 − 2.25i)5-s − 14.7·6-s − 9.33i·7-s + (16.8 + 13.4i)8-s + (6.12 − 2.94i)9-s + (−8.36 − 10.4i)10-s + (0.514 − 0.248i)11-s + (−37.9 − 8.66i)12-s + (−14.4 + 18.1i)13-s + (7.71 − 33.8i)14-s + (12.9 + 6.22i)15-s + (25.4 + 31.9i)16-s + (11.9 + 14.9i)17-s + ⋯
L(s)  = 1  + (1.81 + 0.413i)2-s + (−1.29 + 0.294i)3-s + (2.20 + 1.06i)4-s + (−0.564 − 0.450i)5-s − 2.46·6-s − 1.33i·7-s + (2.10 + 1.67i)8-s + (0.680 − 0.327i)9-s + (−0.836 − 1.04i)10-s + (0.0468 − 0.0225i)11-s + (−3.16 − 0.722i)12-s + (−1.11 + 1.39i)13-s + (0.551 − 2.41i)14-s + (0.862 + 0.415i)15-s + (1.59 + 1.99i)16-s + (0.701 + 0.879i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.846 - 0.532i$
Analytic conductor: \(1.17166\)
Root analytic conductor: \(1.08243\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1),\ 0.846 - 0.532i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73045 + 0.499359i\)
\(L(\frac12)\) \(\approx\) \(1.73045 + 0.499359i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (42.5 + 5.93i)T \)
good2 \( 1 + (-3.62 - 0.826i)T + (3.60 + 1.73i)T^{2} \)
3 \( 1 + (3.87 - 0.884i)T + (8.10 - 3.90i)T^{2} \)
5 \( 1 + (2.82 + 2.25i)T + (5.56 + 24.3i)T^{2} \)
7 \( 1 + 9.33iT - 49T^{2} \)
11 \( 1 + (-0.514 + 0.248i)T + (75.4 - 94.6i)T^{2} \)
13 \( 1 + (14.4 - 18.1i)T + (-37.6 - 164. i)T^{2} \)
17 \( 1 + (-11.9 - 14.9i)T + (-64.3 + 281. i)T^{2} \)
19 \( 1 + (-4.52 + 9.40i)T + (-225. - 282. i)T^{2} \)
23 \( 1 + (2.19 - 1.05i)T + (329. - 413. i)T^{2} \)
29 \( 1 + (-5.29 - 1.20i)T + (757. + 364. i)T^{2} \)
31 \( 1 + (-3.45 + 15.1i)T + (-865. - 416. i)T^{2} \)
37 \( 1 - 35.4iT - 1.36e3T^{2} \)
41 \( 1 + (-6.03 + 26.4i)T + (-1.51e3 - 729. i)T^{2} \)
47 \( 1 + (-39.8 - 19.1i)T + (1.37e3 + 1.72e3i)T^{2} \)
53 \( 1 + (-22.2 - 27.9i)T + (-625. + 2.73e3i)T^{2} \)
59 \( 1 + (17.9 + 22.5i)T + (-774. + 3.39e3i)T^{2} \)
61 \( 1 + (-31.0 + 7.07i)T + (3.35e3 - 1.61e3i)T^{2} \)
67 \( 1 + (66.2 + 31.8i)T + (2.79e3 + 3.50e3i)T^{2} \)
71 \( 1 + (7.23 - 15.0i)T + (-3.14e3 - 3.94e3i)T^{2} \)
73 \( 1 + (102. + 81.7i)T + (1.18e3 + 5.19e3i)T^{2} \)
79 \( 1 + 5.46T + 6.24e3T^{2} \)
83 \( 1 + (-12.2 - 53.5i)T + (-6.20e3 + 2.98e3i)T^{2} \)
89 \( 1 + (-158. + 36.2i)T + (7.13e3 - 3.43e3i)T^{2} \)
97 \( 1 + (73.7 - 35.5i)T + (5.86e3 - 7.35e3i)T^{2} \)
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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

−15.88044124123730203908729342758, −14.56526554658679560217456209581, −13.55659652397122394348529054341, −12.21208130541850488844175284944, −11.67074404471426393392106715790, −10.44047294743616593784171605459, −7.48390201552692430185136550222, −6.38359596527250294930639778191, −4.87112891884484746797891675720, −4.10289919865403001236123772531, 2.95588179216581006950769000405, 5.15345310384211781476256379098, 5.80044531634415019240720075299, 7.28437837114673477305751013753, 10.33013280224060217908547320107, 11.63188868232870508564246877858, 12.04227447521092426444441887287, 12.86019078793541569139016580627, 14.55121476474179021131259268636, 15.30345134083314048275822969936

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