↑

Properties

Label 2-43-43.37-c2-0-4
Degree $2$
Conductor $43$
Sign $-0.622 + 0.782i$
Analytic cond. $1.17166$
Root an. cond. $1.08243$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.51i·2-s + (−3.66 − 2.11i)3-s + 1.71·4-s + (−4.51 − 2.60i)5-s + (−3.20 + 5.54i)6-s + (1.23 − 0.713i)7-s − 8.63i·8-s + (4.47 + 7.75i)9-s + (−3.93 + 6.81i)10-s + 19.5·11-s + (−6.29 − 3.63i)12-s + (4.99 + 8.65i)13-s + (−1.07 − 1.86i)14-s + (11.0 + 19.1i)15-s − 6.19·16-s + (−12.5 − 21.7i)17-s + ⋯
L(s)  = 1  − 0.755i·2-s + (−1.22 − 0.706i)3-s + 0.428·4-s + (−0.902 − 0.520i)5-s + (−0.533 + 0.924i)6-s + (0.176 − 0.101i)7-s − 1.07i·8-s + (0.497 + 0.861i)9-s + (−0.393 + 0.681i)10-s + 1.78·11-s + (−0.524 − 0.302i)12-s + (0.384 + 0.665i)13-s + (−0.0770 − 0.133i)14-s + (0.735 + 1.27i)15-s − 0.387·16-s + (−0.737 − 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.622 + 0.782i$
Analytic conductor: \(1.17166\)
Root analytic conductor: \(1.08243\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1),\ -0.622 + 0.782i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.348661 - 0.722395i\)
\(L(\frac12)\) \(\approx\) \(0.348661 - 0.722395i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-35.8 - 23.7i)T \)
good2 \( 1 + 1.51iT - 4T^{2} \)
3 \( 1 + (3.66 + 2.11i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (4.51 + 2.60i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1.23 + 0.713i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 - 19.5T + 121T^{2} \)
13 \( 1 + (-4.99 - 8.65i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (12.5 + 21.7i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.29 - 5.36i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (16.7 - 28.9i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-13.9 + 8.03i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-3.71 + 6.43i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-18.9 - 10.9i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 22.3T + 1.68e3T^{2} \)
47 \( 1 - 78.3T + 2.20e3T^{2} \)
53 \( 1 + (13.1 - 22.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 39.1T + 3.48e3T^{2} \)
61 \( 1 + (27.5 - 15.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-61.2 + 106. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (110. - 63.6i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-14.9 + 8.61i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (18.2 + 31.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (42.8 - 74.2i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (11.1 + 6.45i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 113.T + 9.40e3T^{2} \)
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

−15.72276378948677614550716929004, −13.82484754075056981276583367450, −12.26766920136914284784895778975, −11.70548011876227645745936539125, −11.28077281682895372767763952180, −9.369870350411390151717125407716, −7.35166944108373304283887689308, −6.27705337983703985648105029504, −4.14792090992533252317070540006, −1.15910080510984721688554023920, 4.12874949726980317289595958301, 5.90782498744645331303729215571, 6.85501472447015163419118969080, 8.480752817064465838809047194163, 10.53527765525260965537607325356, 11.34788935578765739664934637628, 12.12214310543745323117885399215, 14.44235459499319176627529012538, 15.31484957537775186995130159200, 16.08746284561513207397480433355

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