Properties

Label 2-43-43.13-c1-0-2
Degree $2$
Conductor $43$
Sign $-0.000640 + 0.999i$
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  + (0.581 − 2.54i)2-s + (−1.40 + 1.30i)3-s + (−4.34 − 2.09i)4-s + (2.95 + 0.445i)5-s + (2.51 + 4.34i)6-s + (−0.339 + 0.588i)7-s + (−4.59 + 5.76i)8-s + (0.0520 − 0.695i)9-s + (2.85 − 7.26i)10-s + (−1.98 + 0.955i)11-s + (8.86 − 2.73i)12-s + (−0.884 − 2.25i)13-s + (1.30 + 1.20i)14-s + (−4.74 + 3.23i)15-s + (5.99 + 7.52i)16-s + (−2.49 + 0.376i)17-s + ⋯
L(s)  = 1  + (0.411 − 1.80i)2-s + (−0.813 + 0.755i)3-s + (−2.17 − 1.04i)4-s + (1.32 + 0.199i)5-s + (1.02 + 1.77i)6-s + (−0.128 + 0.222i)7-s + (−1.62 + 2.03i)8-s + (0.0173 − 0.231i)9-s + (0.901 − 2.29i)10-s + (−0.597 + 0.287i)11-s + (2.55 − 0.789i)12-s + (−0.245 − 0.624i)13-s + (0.347 + 0.322i)14-s + (−1.22 + 0.835i)15-s + (1.49 + 1.88i)16-s + (−0.606 + 0.0913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000640 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.000640 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.000640 + 0.999i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ -0.000640 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552668 - 0.553022i\)
\(L(\frac12)\) \(\approx\) \(0.552668 - 0.553022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (6.51 + 0.781i)T \)
good2 \( 1 + (-0.581 + 2.54i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (1.40 - 1.30i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (-2.95 - 0.445i)T + (4.77 + 1.47i)T^{2} \)
7 \( 1 + (0.339 - 0.588i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.98 - 0.955i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.884 + 2.25i)T + (-9.52 + 8.84i)T^{2} \)
17 \( 1 + (2.49 - 0.376i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (0.122 + 1.63i)T + (-18.7 + 2.83i)T^{2} \)
23 \( 1 + (-0.0324 - 0.0221i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (-5.57 - 5.17i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (-8.56 + 2.64i)T + (25.6 - 17.4i)T^{2} \)
37 \( 1 + (5.57 + 9.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.555 + 2.43i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-7.92 - 3.81i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (2.23 - 5.70i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-2.97 - 3.73i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (4.87 + 1.50i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (-0.194 - 2.58i)T + (-66.2 + 9.98i)T^{2} \)
71 \( 1 + (7.71 - 5.25i)T + (25.9 - 66.0i)T^{2} \)
73 \( 1 + (1.43 + 3.66i)T + (-53.5 + 49.6i)T^{2} \)
79 \( 1 + (-3.10 + 5.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.97 + 3.68i)T + (6.20 - 82.7i)T^{2} \)
89 \( 1 + (-7.30 + 6.77i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (0.852 - 0.410i)T + (60.4 - 75.8i)T^{2} \)
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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

−15.60207747611093486198313343434, −14.04824055586960595647675060943, −13.15083210585407097089472594247, −12.04921038474262426985226443923, −10.60887980481846860463917074847, −10.33814304124480964463698563687, −9.167175288145413520140400606838, −5.75117227206772712182393661836, −4.71121442403468897439589780461, −2.52830800130209327050015685438, 4.98152333591745475558842390484, 6.15753668809800795525659999643, 6.81122468732977315759772186058, 8.429599088485435052318719117507, 9.904616859131671040670820089412, 12.12071922575939509067082545018, 13.45440848054910383441241513465, 13.74118575081514719795968695705, 15.25959805688474786785476459339, 16.50464287083130484058570418872

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