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Properties

Label 2-43-43.16-c1-0-1
Degree $2$
Conductor $43$
Sign $0.458 - 0.888i$
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 + 2.19i)2-s + (0.346 − 1.51i)3-s + (−2.74 + 1.32i)4-s + (−1.24 − 1.56i)5-s + 3.49·6-s − 4.24·7-s + (−1.46 − 1.84i)8-s + (0.524 + 0.252i)9-s + (2.80 − 3.51i)10-s + (3.96 + 1.91i)11-s + (1.05 + 4.62i)12-s + (0.192 + 0.240i)13-s + (−2.12 − 9.30i)14-s + (−2.80 + 1.34i)15-s + (−0.499 + 0.626i)16-s + (2.80 − 3.51i)17-s + ⋯
L(s)  = 1  + (0.353 + 1.54i)2-s + (0.199 − 0.875i)3-s + (−1.37 + 0.661i)4-s + (−0.557 − 0.699i)5-s + 1.42·6-s − 1.60·7-s + (−0.519 − 0.651i)8-s + (0.174 + 0.0841i)9-s + (0.886 − 1.11i)10-s + (1.19 + 0.576i)11-s + (0.304 + 1.33i)12-s + (0.0532 + 0.0667i)13-s + (−0.567 − 2.48i)14-s + (−0.723 + 0.348i)15-s + (−0.124 + 0.156i)16-s + (0.679 − 0.852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.458 - 0.888i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.458 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.721191 + 0.439608i\)
\(L(\frac12)\) \(\approx\) \(0.721191 + 0.439608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (6.46 - 1.07i)T \)
good2 \( 1 + (-0.5 - 2.19i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-0.346 + 1.51i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (1.24 + 1.56i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 + (-3.96 - 1.91i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-0.192 - 0.240i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-2.80 + 3.51i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + (5.12 - 2.46i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (1.98 + 0.953i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-0.653 - 2.86i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (0.797 + 3.49i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 - 4.87T + 37T^{2} \)
41 \( 1 + (-0.896 - 3.92i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (0.876 - 1.09i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (2.03 - 2.54i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (0.504 - 2.21i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (3.77 - 1.81i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-7.93 + 3.82i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-2.65 - 3.32i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 - 3.71T + 79T^{2} \)
83 \( 1 + (2.26 - 9.90i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (1.11 - 4.87i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (16.4 + 7.91i)T + (60.4 + 75.8i)T^{2} \)
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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

−16.41863640926965833870842133725, −15.18361529081182461952482019397, −13.99027129834562242473952693917, −12.90166588582150157498877668717, −12.25133102619336560860640541073, −9.604403335926798693428507787889, −8.275638814697695049000240753850, −7.06245861639594217538801108413, −6.25884034963225485531291608133, −4.24997829476795876847706237453, 3.31115707884631102767530654690, 3.95829046407918288259505179603, 6.56100686654546890408830000480, 9.119926958165380054166596003623, 10.05450603856824148811021524166, 10.92220312764524339393448868312, 12.17237303203153262332412713684, 13.15761246479429516641632841388, 14.54598812239744521387909657240, 15.63778206950567044259832472412

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