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Properties

Label 2-43-43.6-c3-0-8
Degree $2$
Conductor $43$
Sign $0.146 + 0.989i$
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  + 0.179·2-s + (3.01 − 5.21i)3-s − 7.96·4-s + (5.48 − 9.50i)5-s + (0.541 − 0.937i)6-s + (−4.39 − 7.60i)7-s − 2.87·8-s + (−4.63 − 8.02i)9-s + (0.986 − 1.70i)10-s + 46.3·11-s + (−23.9 + 41.5i)12-s + (−5.68 − 9.84i)13-s + (−0.789 − 1.36i)14-s + (−33.0 − 57.2i)15-s + 63.2·16-s + (25.2 + 43.7i)17-s + ⋯
L(s)  = 1  + 0.0635·2-s + (0.579 − 1.00i)3-s − 0.995·4-s + (0.490 − 0.849i)5-s + (0.0368 − 0.0637i)6-s + (−0.237 − 0.410i)7-s − 0.126·8-s + (−0.171 − 0.297i)9-s + (0.0311 − 0.0540i)10-s + 1.26·11-s + (−0.577 + 0.999i)12-s + (−0.121 − 0.210i)13-s + (−0.0150 − 0.0261i)14-s + (−0.568 − 0.984i)15-s + 0.987·16-s + (0.359 + 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.146 + 0.989i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ 0.146 + 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.06429 - 0.918330i\)
\(L(\frac12)\) \(\approx\) \(1.06429 - 0.918330i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-218. - 177. i)T \)
good2 \( 1 - 0.179T + 8T^{2} \)
3 \( 1 + (-3.01 + 5.21i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-5.48 + 9.50i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (4.39 + 7.60i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 - 46.3T + 1.33e3T^{2} \)
13 \( 1 + (5.68 + 9.84i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-25.2 - 43.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (54.3 - 94.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (27.3 - 47.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (73.2 + 126. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-109. + 188. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-49.7 + 86.1i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 343.T + 6.89e4T^{2} \)
47 \( 1 - 136.T + 1.03e5T^{2} \)
53 \( 1 + (236. - 409. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 626.T + 2.05e5T^{2} \)
61 \( 1 + (-29.9 - 51.9i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (532. - 923. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-553. - 958. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (369. + 640. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (64.6 + 111. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (105. - 183. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (51.4 - 89.1i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 828.T + 9.12e5T^{2} \)
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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

−14.73441734191227018497891895355, −13.78352074478265326807823438041, −13.05795943634249355455965332875, −12.21176521138787201365859605217, −9.963315037737479329966548061070, −8.863937569753955576967724940635, −7.80352476987002821587956615624, −5.99060524894755759425881363177, −4.08759302836675680105045315178, −1.34218641855054392648997981960, 3.22629536757352040382570449772, 4.67388910492322195469959521605, 6.57325122146818006268837683610, 8.824222380626728182898820138670, 9.457285834501632894829665648947, 10.54373875795891765234083792525, 12.27592768218984358870452824588, 13.89403021793330906898255061003, 14.47654477554999356117807402934, 15.40742364595373214390504125665

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