Properties

Label 2-43-43.2-c6-0-6
Degree $2$
Conductor $43$
Sign $0.402 - 0.915i$
Analytic cond. $9.89232$
Root an. cond. $3.14520$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0322 − 0.0669i)2-s + (−17.7 + 36.8i)3-s + (39.8 − 50.0i)4-s + (157. + 36.0i)5-s + 3.03·6-s − 203. i·7-s + (−9.27 − 2.11i)8-s + (−588. − 737. i)9-s + (−2.67 − 11.7i)10-s + (1.39e3 + 1.74e3i)11-s + (1.13e3 + 2.35e3i)12-s + (−91.2 + 399. i)13-s + (−13.6 + 6.55i)14-s + (−4.13e3 + 5.18e3i)15-s + (−911. − 3.99e3i)16-s + (2.06e3 + 9.03e3i)17-s + ⋯
L(s)  = 1  + (−0.00402 − 0.00836i)2-s + (−0.657 + 1.36i)3-s + (0.623 − 0.781i)4-s + (1.26 + 0.288i)5-s + 0.0140·6-s − 0.592i·7-s + (−0.0181 − 0.00413i)8-s + (−0.806 − 1.01i)9-s + (−0.00267 − 0.0117i)10-s + (1.04 + 1.31i)11-s + (0.657 + 1.36i)12-s + (−0.0415 + 0.181i)13-s + (−0.00495 + 0.00238i)14-s + (−1.22 + 1.53i)15-s + (−0.222 − 0.974i)16-s + (0.419 + 1.83i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.402 - 0.915i$
Analytic conductor: \(9.89232\)
Root analytic conductor: \(3.14520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3),\ 0.402 - 0.915i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.65709 + 1.08109i\)
\(L(\frac12)\) \(\approx\) \(1.65709 + 1.08109i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (8.97e3 + 7.89e4i)T \)
good2 \( 1 + (0.0322 + 0.0669i)T + (-39.9 + 50.0i)T^{2} \)
3 \( 1 + (17.7 - 36.8i)T + (-454. - 569. i)T^{2} \)
5 \( 1 + (-157. - 36.0i)T + (1.40e4 + 6.77e3i)T^{2} \)
7 \( 1 + 203. iT - 1.17e5T^{2} \)
11 \( 1 + (-1.39e3 - 1.74e3i)T + (-3.94e5 + 1.72e6i)T^{2} \)
13 \( 1 + (91.2 - 399. i)T + (-4.34e6 - 2.09e6i)T^{2} \)
17 \( 1 + (-2.06e3 - 9.03e3i)T + (-2.17e7 + 1.04e7i)T^{2} \)
19 \( 1 + (-905. - 722. i)T + (1.04e7 + 4.58e7i)T^{2} \)
23 \( 1 + (8.85e3 + 1.11e4i)T + (-3.29e7 + 1.44e8i)T^{2} \)
29 \( 1 + (-1.64e4 - 3.41e4i)T + (-3.70e8 + 4.65e8i)T^{2} \)
31 \( 1 + (-6.78e3 + 3.26e3i)T + (5.53e8 - 6.93e8i)T^{2} \)
37 \( 1 + 3.95e4iT - 2.56e9T^{2} \)
41 \( 1 + (-9.89e4 + 4.76e4i)T + (2.96e9 - 3.71e9i)T^{2} \)
47 \( 1 + (8.60e4 - 1.07e5i)T + (-2.39e9 - 1.05e10i)T^{2} \)
53 \( 1 + (4.48e4 + 1.96e5i)T + (-1.99e10 + 9.61e9i)T^{2} \)
59 \( 1 + (-3.47e3 - 1.52e4i)T + (-3.80e10 + 1.83e10i)T^{2} \)
61 \( 1 + (3.83e4 - 7.95e4i)T + (-3.21e10 - 4.02e10i)T^{2} \)
67 \( 1 + (-3.99e4 + 5.01e4i)T + (-2.01e10 - 8.81e10i)T^{2} \)
71 \( 1 + (1.60e5 + 1.27e5i)T + (2.85e10 + 1.24e11i)T^{2} \)
73 \( 1 + (-4.64e4 - 1.06e4i)T + (1.36e11 + 6.56e10i)T^{2} \)
79 \( 1 - 1.36e3T + 2.43e11T^{2} \)
83 \( 1 + (2.47e5 + 1.19e5i)T + (2.03e11 + 2.55e11i)T^{2} \)
89 \( 1 + (-8.35e4 + 1.73e5i)T + (-3.09e11 - 3.88e11i)T^{2} \)
97 \( 1 + (-2.11e5 - 2.65e5i)T + (-1.85e11 + 8.12e11i)T^{2} \)
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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

−14.77325468177438892800176377569, −14.33207136395945094460513883295, −12.32048451308824677227288103082, −10.72026096536617647966010942069, −10.24876875902225681190077282669, −9.431320590385765982009968390881, −6.68341944063083453507854452545, −5.69961224968650677878901139537, −4.23456030200113633425588524331, −1.72291275501263543367582956707, 1.14711245767157024706601369693, 2.64243283730603522666136723918, 5.75135380891367364999552231764, 6.50651044874899340178113489105, 7.915598976625445385098827980988, 9.399853080754380314321348431615, 11.53321207109688322361743390584, 11.95097554192275229145554523501, 13.28988642506745586698326127753, 13.91743479825056297231704124056

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