Properties

Label 2-43-43.38-c1-0-1
Degree $2$
Conductor $43$
Sign $-0.00542 - 0.999i$
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  + (−0.515 + 2.25i)2-s + (1.28 + 0.397i)3-s + (−3.03 − 1.46i)4-s + (−1.48 − 3.79i)5-s + (−1.56 + 2.70i)6-s + (1.38 + 2.40i)7-s + (1.98 − 2.48i)8-s + (−0.976 − 0.665i)9-s + (9.33 − 1.40i)10-s + (−0.678 + 0.326i)11-s + (−3.33 − 3.09i)12-s + (1.70 + 0.256i)13-s + (−6.15 + 1.89i)14-s + (−0.410 − 5.47i)15-s + (0.394 + 0.495i)16-s + (−1.18 + 3.02i)17-s + ⋯
L(s)  = 1  + (−0.364 + 1.59i)2-s + (0.743 + 0.229i)3-s + (−1.51 − 0.731i)4-s + (−0.665 − 1.69i)5-s + (−0.638 + 1.10i)6-s + (0.525 + 0.909i)7-s + (0.701 − 0.880i)8-s + (−0.325 − 0.221i)9-s + (2.95 − 0.444i)10-s + (−0.204 + 0.0984i)11-s + (−0.962 − 0.893i)12-s + (0.471 + 0.0711i)13-s + (−1.64 + 0.507i)14-s + (−0.105 − 1.41i)15-s + (0.0987 + 0.123i)16-s + (−0.288 + 0.733i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00542 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00542 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.00542 - 0.999i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ -0.00542 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509843 + 0.512615i\)
\(L(\frac12)\) \(\approx\) \(0.509843 + 0.512615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (4.39 - 4.86i)T \)
good2 \( 1 + (0.515 - 2.25i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (-1.28 - 0.397i)T + (2.47 + 1.68i)T^{2} \)
5 \( 1 + (1.48 + 3.79i)T + (-3.66 + 3.40i)T^{2} \)
7 \( 1 + (-1.38 - 2.40i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.678 - 0.326i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-1.70 - 0.256i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (1.18 - 3.02i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (0.0395 - 0.0269i)T + (6.94 - 17.6i)T^{2} \)
23 \( 1 + (-0.152 + 2.03i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (-0.714 + 0.220i)T + (23.9 - 16.3i)T^{2} \)
31 \( 1 + (-2.52 - 2.34i)T + (2.31 + 30.9i)T^{2} \)
37 \( 1 + (-3.91 + 6.78i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.86 - 8.18i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-7.05 - 3.39i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (1.76 - 0.266i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-3.60 - 4.52i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-2.15 + 2.00i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-4.62 + 3.15i)T + (24.4 - 62.3i)T^{2} \)
71 \( 1 + (0.543 + 7.25i)T + (-70.2 + 10.5i)T^{2} \)
73 \( 1 + (10.0 + 1.51i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (6.70 + 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.42 - 1.67i)T + (68.5 + 46.7i)T^{2} \)
89 \( 1 + (11.7 + 3.63i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 + (-9.41 + 4.53i)T + (60.4 - 75.8i)T^{2} \)
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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

−16.09492097411798672290694370569, −15.37332870716813948951312146312, −14.54580701449812175755114584836, −13.12845825632182508224810479116, −11.81278315719740428739265113230, −9.246871461920392019630365218574, −8.561906631418509240061824006573, −8.006858191563632491574543019104, −5.86044460735142055589250032045, −4.55786041807976258603431588350, 2.62169975227328129946303761303, 3.78397421188759006014154895343, 7.19811260490917150647193296520, 8.383229297770307344424210550922, 10.10036718893538276794419924527, 11.01085926289056121856475235927, 11.62700445548754781977736602363, 13.49845910836832937826256216679, 14.13116907920716859281237995528, 15.40905054977290880895178070201

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