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Properties

Label 2-43-43.16-c1-0-0
Degree $2$
Conductor $43$
Sign $0.997 - 0.0667i$
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  + (−0.0990 − 0.433i)2-s + (−0.400 + 1.75i)3-s + (1.62 − 0.781i)4-s + (−0.5 − 0.626i)5-s + 0.801·6-s − 3.04·7-s + (−1.05 − 1.32i)8-s + (−0.222 − 0.107i)9-s + (−0.222 + 0.279i)10-s + (−4.77 − 2.29i)11-s + (0.722 + 3.16i)12-s + (2.46 + 3.09i)13-s + (0.301 + 1.32i)14-s + (1.30 − 0.626i)15-s + (1.77 − 2.22i)16-s + (−0.554 + 0.695i)17-s + ⋯
L(s)  = 1  + (−0.0700 − 0.306i)2-s + (−0.231 + 1.01i)3-s + (0.811 − 0.390i)4-s + (−0.223 − 0.280i)5-s + 0.327·6-s − 1.15·7-s + (−0.372 − 0.467i)8-s + (−0.0741 − 0.0357i)9-s + (−0.0703 + 0.0882i)10-s + (−1.43 − 0.692i)11-s + (0.208 + 0.913i)12-s + (0.684 + 0.858i)13-s + (0.0806 + 0.353i)14-s + (0.336 − 0.161i)15-s + (0.444 − 0.557i)16-s + (−0.134 + 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.997 - 0.0667i$
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: \((2,\ 43,\ (\ :1/2),\ 0.997 - 0.0667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.751948 + 0.0251144i\)
\(L(\frac12)\) \(\approx\) \(0.751948 + 0.0251144i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (4.25 + 4.98i)T \)
good2 \( 1 + (0.0990 + 0.433i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (0.400 - 1.75i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (0.5 + 0.626i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + 3.04T + 7T^{2} \)
11 \( 1 + (4.77 + 2.29i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-2.46 - 3.09i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.554 - 0.695i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + (-2.48 + 1.19i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-3.94 - 1.90i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-1.33 - 5.84i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-1.54 - 6.77i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + 3.46T + 37T^{2} \)
41 \( 1 + (1.58 + 6.93i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (-8.02 + 3.86i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.29 - 1.61i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (0.538 - 0.674i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-0.307 + 1.34i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (6.04 - 2.91i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-14.9 + 7.19i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (7.76 + 9.73i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + 7.85T + 79T^{2} \)
83 \( 1 + (1.63 - 7.18i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (3.13 - 13.7i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-8.44 - 4.06i)T + (60.4 + 75.8i)T^{2} \)
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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

−15.96513469488941654167347733261, −15.48340996772150848748719332338, −13.64254293194229275303667612863, −12.32713277486110619684327314502, −10.88853351163136093895288478736, −10.28816309491343505100715689064, −8.971944856428786149472239688581, −6.88255250441140821997469908598, −5.32565977658481425889969363294, −3.30347768089725826481993777647, 2.89561549258642634439985197996, 5.97748392060107263565624987378, 7.10419038033611714736433613675, 7.941318874772606490919485943225, 10.05962326654539018714442984786, 11.43440138764380748587667695008, 12.75718564275894367007927907457, 13.16696623169844818459288871260, 15.31461508508956173874536317902, 15.81245531460919198786925062425

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