Properties

Label 2-43-43.12-c8-0-26
Degree $2$
Conductor $43$
Sign $-0.731 - 0.681i$
Analytic cond. $17.5172$
Root an. cond. $4.18536$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.46 − 7.18i)2-s + (−10.4 − 15.3i)3-s + (119. − 150. i)4-s + (61.3 − 66.1i)5-s + (−74.1 + 128. i)6-s + (−2.36e3 + 1.36e3i)7-s + (−3.48e3 − 796. i)8-s + (2.27e3 − 5.78e3i)9-s + (−687. − 212. i)10-s + (−2.24e3 − 2.81e3i)11-s + (−3.56e3 − 267. i)12-s + (1.68e4 − 5.19e3i)13-s + (1.80e4 + 1.22e4i)14-s + (−1.65e3 − 249. i)15-s + (−4.60e3 − 2.01e4i)16-s + (−4.61e4 + 4.27e4i)17-s + ⋯
L(s)  = 1  + (−0.216 − 0.449i)2-s + (−0.129 − 0.189i)3-s + (0.468 − 0.587i)4-s + (0.0981 − 0.105i)5-s + (−0.0572 + 0.0990i)6-s + (−0.986 + 0.569i)7-s + (−0.851 − 0.194i)8-s + (0.346 − 0.881i)9-s + (−0.0687 − 0.0212i)10-s + (−0.153 − 0.192i)11-s + (−0.171 − 0.0128i)12-s + (0.589 − 0.181i)13-s + (0.469 + 0.320i)14-s + (−0.0327 − 0.00493i)15-s + (−0.0702 − 0.307i)16-s + (−0.552 + 0.512i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.731 - 0.681i$
Analytic conductor: \(17.5172\)
Root analytic conductor: \(4.18536\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :4),\ -0.731 - 0.681i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.136475 + 0.346646i\)
\(L(\frac12)\) \(\approx\) \(0.136475 + 0.346646i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (2.84e6 + 1.90e6i)T \)
good2 \( 1 + (3.46 + 7.18i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (10.4 + 15.3i)T + (-2.39e3 + 6.10e3i)T^{2} \)
5 \( 1 + (-61.3 + 66.1i)T + (-2.91e4 - 3.89e5i)T^{2} \)
7 \( 1 + (2.36e3 - 1.36e3i)T + (2.88e6 - 4.99e6i)T^{2} \)
11 \( 1 + (2.24e3 + 2.81e3i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (-1.68e4 + 5.19e3i)T + (6.73e8 - 4.59e8i)T^{2} \)
17 \( 1 + (4.61e4 - 4.27e4i)T + (5.21e8 - 6.95e9i)T^{2} \)
19 \( 1 + (1.75e5 - 6.89e4i)T + (1.24e10 - 1.15e10i)T^{2} \)
23 \( 1 + (9.30e4 - 1.40e4i)T + (7.48e10 - 2.30e10i)T^{2} \)
29 \( 1 + (4.03e5 - 5.92e5i)T + (-1.82e11 - 4.65e11i)T^{2} \)
31 \( 1 + (-3.83e4 + 5.12e5i)T + (-8.43e11 - 1.27e11i)T^{2} \)
37 \( 1 + (-8.16e5 - 4.71e5i)T + (1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + (1.80e6 - 8.71e5i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (-3.51e6 + 4.41e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (-2.96e6 - 9.13e5i)T + (5.14e13 + 3.50e13i)T^{2} \)
59 \( 1 + (4.38e6 + 1.92e7i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (-1.56e6 + 1.16e5i)T + (1.89e14 - 2.85e13i)T^{2} \)
67 \( 1 + (6.91e6 + 1.76e7i)T + (-2.97e14 + 2.76e14i)T^{2} \)
71 \( 1 + (-4.71e6 + 3.12e7i)T + (-6.17e14 - 1.90e14i)T^{2} \)
73 \( 1 + (-6.23e6 - 2.02e7i)T + (-6.66e14 + 4.54e14i)T^{2} \)
79 \( 1 + (1.24e7 + 2.15e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + (2.83e7 - 1.93e7i)T + (8.22e14 - 2.09e15i)T^{2} \)
89 \( 1 + (3.68e7 + 5.40e7i)T + (-1.43e15 + 3.66e15i)T^{2} \)
97 \( 1 + (-8.20e6 - 1.02e7i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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

−13.12961354555787508225062514882, −12.28716167290939352798577332771, −11.01368771444840047818346112599, −9.857621553594591263119706787434, −8.811406116459344494375831103271, −6.65612940677137806438265803951, −5.86681619869788496295476269824, −3.45763846027922655571579165693, −1.77729617226675896185277516406, −0.14146293358255038954885719153, 2.46947293177832320202454116095, 4.18234810428310663354111393763, 6.25633951255088991349647336744, 7.24544961025746841898544796995, 8.601549272394377765668509069326, 10.12967689309030577004744364221, 11.23397658672542531832044612501, 12.73544291136798320877346227927, 13.62318087029290063751810526644, 15.32871860416174400646871697745

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