Properties

Label 2-43-43.4-c1-0-1
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: $\chi_{43} (4, \cdot )$
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

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

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