Properties

Label 2-43-43.13-c1-0-1
Degree $2$
Conductor $43$
Sign $0.857 + 0.514i$
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  + (0.188 − 0.826i)2-s + (−0.0129 + 0.0119i)3-s + (1.15 + 0.556i)4-s + (−3.39 − 0.512i)5-s + (0.00747 + 0.0129i)6-s + (−0.134 + 0.232i)7-s + (1.73 − 2.17i)8-s + (−0.224 + 2.99i)9-s + (−1.06 + 2.71i)10-s + (−2.96 + 1.42i)11-s + (−0.0216 + 0.00666i)12-s + (−0.736 − 1.87i)13-s + (0.167 + 0.155i)14-s + (0.0500 − 0.0341i)15-s + (0.129 + 0.161i)16-s + (6.37 − 0.960i)17-s + ⋯
L(s)  = 1  + (0.133 − 0.584i)2-s + (−0.00746 + 0.00692i)3-s + (0.577 + 0.278i)4-s + (−1.51 − 0.229i)5-s + (0.00305 + 0.00528i)6-s + (−0.0508 + 0.0880i)7-s + (0.613 − 0.768i)8-s + (−0.0747 + 0.997i)9-s + (−0.336 + 0.857i)10-s + (−0.894 + 0.430i)11-s + (−0.00623 + 0.00192i)12-s + (−0.204 − 0.520i)13-s + (0.0446 + 0.0414i)14-s + (0.0129 − 0.00881i)15-s + (0.0322 + 0.0404i)16-s + (1.54 − 0.233i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.857 + 0.514i$
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: \((2,\ 43,\ (\ :1/2),\ 0.857 + 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.769080 - 0.213138i\)
\(L(\frac12)\) \(\approx\) \(0.769080 - 0.213138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (4.01 - 5.18i)T \)
good2 \( 1 + (-0.188 + 0.826i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.0129 - 0.0119i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (3.39 + 0.512i)T + (4.77 + 1.47i)T^{2} \)
7 \( 1 + (0.134 - 0.232i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.96 - 1.42i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.736 + 1.87i)T + (-9.52 + 8.84i)T^{2} \)
17 \( 1 + (-6.37 + 0.960i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (0.449 + 5.99i)T + (-18.7 + 2.83i)T^{2} \)
23 \( 1 + (1.83 + 1.25i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (-1.75 - 1.63i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (4.93 - 1.52i)T + (25.6 - 17.4i)T^{2} \)
37 \( 1 + (-2.63 - 4.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.643 - 2.81i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (5.31 + 2.56i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-2.34 + 5.97i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-8.36 - 10.4i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (9.50 + 2.93i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (0.950 + 12.6i)T + (-66.2 + 9.98i)T^{2} \)
71 \( 1 + (4.98 - 3.40i)T + (25.9 - 66.0i)T^{2} \)
73 \( 1 + (-0.609 - 1.55i)T + (-53.5 + 49.6i)T^{2} \)
79 \( 1 + (-6.97 + 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.60 - 5.19i)T + (6.20 - 82.7i)T^{2} \)
89 \( 1 + (1.32 - 1.23i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (-2.36 + 1.13i)T + (60.4 - 75.8i)T^{2} \)
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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

−15.97485525950707828691481746814, −15.03207855865339642475973709692, −13.17173066950531623174649144725, −12.24462184449001221333277607918, −11.29664459352253528223080781012, −10.28169892285607282831575736646, −8.064465185726210142267160785387, −7.37618891774067609623104052879, −4.82003467776036744184842805854, −3.02817159373934181191224135386, 3.64735500018803253957244067901, 5.76965005998965698879881645177, 7.29092140710799427239364958072, 8.152075516225040642416897040102, 10.23266760642578425059424261129, 11.52722667411031890609602540181, 12.32374161895736394167348059426, 14.32723998236206583005643924187, 15.05304986242327805311567279648, 16.01187373706816386090014790560

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