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Properties

Label 2-43-43.27-c2-0-6
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: $\chi_{43} (27, \cdot )$
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

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

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