↑

Properties

Label 2-43-43.41-c1-0-1
Degree $2$
Conductor $43$
Sign $0.996 + 0.0789i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.240i)2-s + (0.178 + 0.0859i)3-s + (−1.05 + 1.32i)4-s + (0.445 − 1.94i)5-s + 0.109·6-s − 2.55·7-s + (−0.455 + 1.99i)8-s + (−1.84 − 2.31i)9-s + (−0.246 − 1.08i)10-s + (2.95 + 3.70i)11-s + (−0.301 + 0.145i)12-s + (−0.143 + 0.626i)13-s + (−1.27 + 0.615i)14-s + (0.246 − 0.309i)15-s + (−0.500 − 2.19i)16-s + (−0.246 − 1.08i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.170i)2-s + (0.103 + 0.0496i)3-s + (−0.527 + 0.661i)4-s + (0.199 − 0.872i)5-s + 0.0448·6-s − 0.965·7-s + (−0.161 + 0.706i)8-s + (−0.615 − 0.771i)9-s + (−0.0781 − 0.342i)10-s + (0.891 + 1.11i)11-s + (−0.0871 + 0.0419i)12-s + (−0.0396 + 0.173i)13-s + (−0.341 + 0.164i)14-s + (0.0637 − 0.0799i)15-s + (−0.125 − 0.547i)16-s + (−0.0599 − 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.996 + 0.0789i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.996 + 0.0789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.836284 - 0.0330587i\)
\(L(\frac12)\) \(\approx\) \(0.836284 - 0.0330587i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (5.45 - 3.63i)T \)
good2 \( 1 + (-0.5 + 0.240i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (-0.178 - 0.0859i)T + (1.87 + 2.34i)T^{2} \)
5 \( 1 + (-0.445 + 1.94i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + 2.55T + 7T^{2} \)
11 \( 1 + (-2.95 - 3.70i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (0.143 - 0.626i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (0.246 + 1.08i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + (-3.33 + 4.18i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-3.27 - 4.10i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-0.821 + 0.395i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (5.37 - 2.58i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + (-6.99 + 3.36i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (0.623 - 0.781i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (1.72 + 7.54i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-1.85 - 8.12i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-7.11 - 3.42i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (-0.291 + 0.364i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-5.91 + 7.41i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-2.82 + 12.3i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + 5.09T + 79T^{2} \)
83 \( 1 + (13.4 + 6.47i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-15.3 - 7.38i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-10.3 - 12.9i)T + (-21.5 + 94.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.10148772697550860910506054052, −14.68167199433446833216270308425, −13.45183587899407524681287726772, −12.54912843581988426019599765062, −11.73766216422731739604462584160, −9.396344263823800360130217675790, −9.002748540916390794843037881396, −7.00385334769351368487781017863, −5.06587383033595705286424851821, −3.45672301403236084943702316106, 3.40179336612502443232279256047, 5.62629719788149797121490619666, 6.68470187212503501191963663146, 8.709266190012255589339118589365, 10.03487273941397967342955116996, 11.08358939721922990004017426802, 12.84351815882347094861860630719, 14.06510352917959927194420407864, 14.44275034862139380622590988641, 15.93672698548587131640098840640

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