Properties

Label 2-43-43.40-c1-0-2
Degree $2$
Conductor $43$
Sign $0.411 + 0.911i$
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  + (−0.982 − 0.473i)2-s + (0.109 − 1.45i)3-s + (−0.504 − 0.633i)4-s + (1.29 + 0.398i)5-s + (−0.796 + 1.37i)6-s + (−0.108 − 0.187i)7-s + (0.682 + 2.98i)8-s + (0.860 + 0.129i)9-s + (−1.08 − 1.00i)10-s + (−3.76 + 4.71i)11-s + (−0.976 + 0.665i)12-s + (2.10 − 1.95i)13-s + (0.0176 + 0.235i)14-s + (0.720 − 1.83i)15-s + (0.383 − 1.68i)16-s + (0.270 − 0.0833i)17-s + ⋯
L(s)  = 1  + (−0.695 − 0.334i)2-s + (0.0629 − 0.840i)3-s + (−0.252 − 0.316i)4-s + (0.577 + 0.178i)5-s + (−0.325 + 0.562i)6-s + (−0.0408 − 0.0708i)7-s + (0.241 + 1.05i)8-s + (0.286 + 0.0432i)9-s + (−0.341 − 0.316i)10-s + (−1.13 + 1.42i)11-s + (−0.281 + 0.192i)12-s + (0.584 − 0.542i)13-s + (0.00471 + 0.0629i)14-s + (0.185 − 0.473i)15-s + (0.0959 − 0.420i)16-s + (0.0654 − 0.0202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.411 + 0.911i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.411 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.499348 - 0.322334i\)
\(L(\frac12)\) \(\approx\) \(0.499348 - 0.322334i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-5.90 + 2.85i)T \)
good2 \( 1 + (0.982 + 0.473i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (-0.109 + 1.45i)T + (-2.96 - 0.447i)T^{2} \)
5 \( 1 + (-1.29 - 0.398i)T + (4.13 + 2.81i)T^{2} \)
7 \( 1 + (0.108 + 0.187i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.76 - 4.71i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-2.10 + 1.95i)T + (0.971 - 12.9i)T^{2} \)
17 \( 1 + (-0.270 + 0.0833i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-1.12 + 0.169i)T + (18.1 - 5.60i)T^{2} \)
23 \( 1 + (1.44 + 3.67i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (-0.515 - 6.88i)T + (-28.6 + 4.32i)T^{2} \)
31 \( 1 + (8.17 - 5.57i)T + (11.3 - 28.8i)T^{2} \)
37 \( 1 + (3.77 - 6.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.62 + 2.22i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (-0.288 - 0.361i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (6.12 + 5.68i)T + (3.96 + 52.8i)T^{2} \)
59 \( 1 + (-1.85 + 8.14i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (-11.0 - 7.56i)T + (22.2 + 56.7i)T^{2} \)
67 \( 1 + (-6.22 + 0.938i)T + (64.0 - 19.7i)T^{2} \)
71 \( 1 + (0.540 - 1.37i)T + (-52.0 - 48.2i)T^{2} \)
73 \( 1 + (-0.601 + 0.557i)T + (5.45 - 72.7i)T^{2} \)
79 \( 1 + (3.07 + 5.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.538 - 7.18i)T + (-82.0 - 12.3i)T^{2} \)
89 \( 1 + (-0.331 + 4.42i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (2.78 - 3.49i)T + (-21.5 - 94.5i)T^{2} \)
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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

−15.84895750079935984728481795602, −14.47345613565719722999358922732, −13.34974710753938436707412617679, −12.43606751635015046336128264534, −10.58684126308794539867210721150, −9.916390160612357466343641284771, −8.345007027817108548039303057863, −7.04183416608363028040395053365, −5.22612001412813298720435669375, −1.95501408328108238800113214660, 3.80393833045568166050856234869, 5.71852402652614412247427193879, 7.70460807350004258772833565788, 8.996181798707910439650829157117, 9.835688905428098912897863751657, 11.08886189758593039526850589438, 12.99301926223276711372145960414, 13.81415839453158211239535836681, 15.65323093213403261781778616571, 16.17312455438772815520426825272

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