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Properties

Label 2-43-43.35-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: $\chi_{43} (35, \cdot )$
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

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

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