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Properties

Label 2-43-43.4-c1-0-0
Degree $2$
Conductor $43$
Sign $0.469 - 0.883i$
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  + (−0.777 + 0.974i)2-s + (0.277 + 0.347i)3-s + (0.0990 + 0.433i)4-s + (−0.5 − 0.240i)5-s − 0.554·6-s + 1.35·7-s + (−2.74 − 1.32i)8-s + (0.623 − 2.73i)9-s + (0.623 − 0.300i)10-s + (0.480 − 2.10i)11-s + (−0.123 + 0.154i)12-s + (−1.42 − 0.686i)13-s + (−1.05 + 1.32i)14-s + (−0.0549 − 0.240i)15-s + (2.62 − 1.26i)16-s + (−2.24 + 1.08i)17-s + ⋯
L(s)  = 1  + (−0.549 + 0.689i)2-s + (0.160 + 0.200i)3-s + (0.0495 + 0.216i)4-s + (−0.223 − 0.107i)5-s − 0.226·6-s + 0.512·7-s + (−0.971 − 0.467i)8-s + (0.207 − 0.910i)9-s + (0.197 − 0.0949i)10-s + (0.144 − 0.634i)11-s + (−0.0356 + 0.0447i)12-s + (−0.395 − 0.190i)13-s + (−0.281 + 0.353i)14-s + (−0.0141 − 0.0621i)15-s + (0.655 − 0.315i)16-s + (−0.544 + 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.469 - 0.883i$
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: \((2,\ 43,\ (\ :1/2),\ 0.469 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557150 + 0.334954i\)
\(L(\frac12)\) \(\approx\) \(0.557150 + 0.334954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (3.91 - 5.26i)T \)
good2 \( 1 + (0.777 - 0.974i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.277 - 0.347i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (0.5 + 0.240i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 - 1.35T + 7T^{2} \)
11 \( 1 + (-0.480 + 2.10i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.42 + 0.686i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (2.24 - 1.08i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + (-1.29 - 5.65i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (1.13 - 4.97i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (3.74 - 4.69i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-4.75 + 5.96i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + 7.18T + 37T^{2} \)
41 \( 1 + (-6.54 + 8.20i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-2.43 - 10.6i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-2.77 + 1.33i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-6.57 + 3.16i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-5.04 - 6.33i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (1.64 + 7.19i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-0.374 - 1.64i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-3.58 - 1.72i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + 2.08T + 79T^{2} \)
83 \( 1 + (-0.571 - 0.717i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (0.928 + 1.16i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-3.36 + 14.7i)T + (-87.3 - 42.0i)T^{2} \)
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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

−16.16397315482699533536219482441, −15.34079133234003268477918184890, −14.25198924075266258208483195521, −12.57474481135120933157053071481, −11.56547043220731755354190298859, −9.786896569700266012538204427237, −8.606508672414175065369680913843, −7.56226390010543175341079984875, −6.04124441270756831346492612690, −3.73060713755897409966563918077, 2.22392575225500720392722814358, 4.93311376787048380702474072068, 7.03917014221067456600053430680, 8.539075099182556760395071527711, 9.867927823822827493076867862177, 10.99516848689308446591231760875, 11.92822257880927123777479302521, 13.45208144455097362504799681396, 14.68827430697658238776667310948, 15.70636317918828612916238266559

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