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Properties

Label 2-43-43.11-c1-0-0
Degree $2$
Conductor $43$
Sign $0.526 - 0.849i$
Analytic cond. $0.343356$
Root an. cond. $0.585966$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.626i)2-s + (−2.02 + 2.53i)3-s + (0.301 − 1.32i)4-s + (1.80 − 0.867i)5-s − 2.60·6-s − 1.19·7-s + (2.42 − 1.16i)8-s + (−1.67 − 7.35i)9-s + (1.44 + 0.695i)10-s + (0.0745 + 0.326i)11-s + (2.74 + 3.44i)12-s + (−4.54 + 2.19i)13-s + (−0.599 − 0.751i)14-s + (−1.44 + 6.33i)15-s + (−0.500 − 0.240i)16-s + (1.44 + 0.695i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.443i)2-s + (−1.16 + 1.46i)3-s + (0.150 − 0.661i)4-s + (0.805 − 0.388i)5-s − 1.06·6-s − 0.452·7-s + (0.857 − 0.412i)8-s + (−0.559 − 2.45i)9-s + (0.456 + 0.220i)10-s + (0.0224 + 0.0985i)11-s + (0.792 + 0.994i)12-s + (−1.26 + 0.607i)13-s + (−0.160 − 0.200i)14-s + (−0.373 + 1.63i)15-s + (−0.125 − 0.0601i)16-s + (0.350 + 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.526 - 0.849i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.526 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.660856 + 0.367842i\)
\(L(\frac12)\) \(\approx\) \(0.660856 + 0.367842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (2.57 - 6.03i)T \)
good2 \( 1 + (-0.5 - 0.626i)T + (-0.445 + 1.94i)T^{2} \)
3 \( 1 + (2.02 - 2.53i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (-1.80 + 0.867i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + 1.19T + 7T^{2} \)
11 \( 1 + (-0.0745 - 0.326i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (4.54 - 2.19i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-1.44 - 0.695i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 + (0.211 - 0.927i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (0.791 + 3.46i)T + (-20.7 + 9.97i)T^{2} \)
29 \( 1 + (-3.02 - 3.79i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (2.83 + 3.55i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 - 4.52T + 37T^{2} \)
41 \( 1 + (-3.60 - 4.52i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (-0.222 + 0.974i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (2.40 + 1.15i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (-11.1 - 5.38i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-2.88 + 3.62i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (-1.48 + 6.48i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-0.149 + 0.653i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.02 + 2.41i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 - 4.38T + 79T^{2} \)
83 \( 1 + (3.79 - 4.75i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (8.71 - 10.9i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (3.38 + 14.8i)T + (-87.3 + 42.0i)T^{2} \)
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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

−16.45918144196037285880735873668, −15.13551451679858443164819193393, −14.34911186104051247236347146037, −12.63041125209460759221240379404, −11.21508083697656252559891305160, −9.946143452517679062127432810558, −9.574222191553289821899651624354, −6.54268526852610464752746187965, −5.51002429435433884902934957058, −4.52120400793101229224719822202, 2.41100834488408980377657389751, 5.40150888014328028533476114716, 6.76550212258728079683357292888, 7.77964484501620124659092777808, 10.20568498198410966420046612367, 11.48392275785048907969023689350, 12.38629825907513780393835546089, 13.12465422889023663995955514645, 14.06566251002530967078148214439, 16.25717665816097229992301962720

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