Properties

Label 2-43-43.2-c8-0-15
Degree $2$
Conductor $43$
Sign $-0.418 + 0.908i$
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  + (−5.04 − 10.4i)2-s + (−24.0 + 49.9i)3-s + (75.4 − 94.6i)4-s + (11.0 + 2.51i)5-s + 643.·6-s − 674. i·7-s + (−4.27e3 − 974. i)8-s + (2.17e3 + 2.73e3i)9-s + (−29.2 − 128. i)10-s + (1.13e4 + 1.42e4i)11-s + (2.90e3 + 6.04e3i)12-s + (7.86e3 − 3.44e4i)13-s + (−7.06e3 + 3.40e3i)14-s + (−391. + 490. i)15-s + (4.42e3 + 1.93e4i)16-s + (−2.81e4 − 1.23e5i)17-s + ⋯
L(s)  = 1  + (−0.315 − 0.654i)2-s + (−0.296 + 0.616i)3-s + (0.294 − 0.369i)4-s + (0.0176 + 0.00403i)5-s + 0.496·6-s − 0.281i·7-s + (−1.04 − 0.237i)8-s + (0.331 + 0.416i)9-s + (−0.00292 − 0.0128i)10-s + (0.776 + 0.973i)11-s + (0.140 + 0.291i)12-s + (0.275 − 1.20i)13-s + (−0.183 + 0.0885i)14-s + (−0.00772 + 0.00968i)15-s + (0.0675 + 0.295i)16-s + (−0.336 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.418 + 0.908i$
Analytic conductor: \(17.5172\)
Root analytic conductor: \(4.18536\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :4),\ -0.418 + 0.908i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.694441 - 1.08519i\)
\(L(\frac12)\) \(\approx\) \(0.694441 - 1.08519i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (3.26e5 + 3.40e6i)T \)
good2 \( 1 + (5.04 + 10.4i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (24.0 - 49.9i)T + (-4.09e3 - 5.12e3i)T^{2} \)
5 \( 1 + (-11.0 - 2.51i)T + (3.51e5 + 1.69e5i)T^{2} \)
7 \( 1 + 674. iT - 5.76e6T^{2} \)
11 \( 1 + (-1.13e4 - 1.42e4i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (-7.86e3 + 3.44e4i)T + (-7.34e8 - 3.53e8i)T^{2} \)
17 \( 1 + (2.81e4 + 1.23e5i)T + (-6.28e9 + 3.02e9i)T^{2} \)
19 \( 1 + (-3.94e4 - 3.14e4i)T + (3.77e9 + 1.65e10i)T^{2} \)
23 \( 1 + (1.13e5 + 1.41e5i)T + (-1.74e10 + 7.63e10i)T^{2} \)
29 \( 1 + (4.53e5 + 9.41e5i)T + (-3.11e11 + 3.91e11i)T^{2} \)
31 \( 1 + (-1.19e6 + 5.75e5i)T + (5.31e11 - 6.66e11i)T^{2} \)
37 \( 1 + 2.38e6iT - 3.51e12T^{2} \)
41 \( 1 + (-3.16e6 + 1.52e6i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (3.57e6 - 4.48e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (-1.78e6 - 7.80e6i)T + (-5.60e13 + 2.70e13i)T^{2} \)
59 \( 1 + (1.78e6 + 7.82e6i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (8.28e6 - 1.72e7i)T + (-1.19e14 - 1.49e14i)T^{2} \)
67 \( 1 + (-6.68e6 + 8.38e6i)T + (-9.03e13 - 3.95e14i)T^{2} \)
71 \( 1 + (7.24e5 + 5.77e5i)T + (1.43e14 + 6.29e14i)T^{2} \)
73 \( 1 + (-3.06e7 - 7.00e6i)T + (7.26e14 + 3.49e14i)T^{2} \)
79 \( 1 + 6.41e7T + 1.51e15T^{2} \)
83 \( 1 + (-6.76e7 - 3.25e7i)T + (1.40e15 + 1.76e15i)T^{2} \)
89 \( 1 + (-7.71e6 + 1.60e7i)T + (-2.45e15 - 3.07e15i)T^{2} \)
97 \( 1 + (9.66e7 + 1.21e8i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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

−13.83182898567846439211437027224, −12.28101625623635663028494421103, −11.22935356532954169358801477395, −10.16270912856075253831108326712, −9.491660521184907605068755866648, −7.47973600821305810877754131851, −5.82165369567962752831819088729, −4.22983074046314820062839612627, −2.30405753394251201606937767695, −0.57864050863752678649749177930, 1.51869107713564945010829437681, 3.65125484563133613231779130006, 6.10142538674590924973572272457, 6.75764003279121130162511201124, 8.214562723398092437443046830593, 9.290701755040632044346642225346, 11.35598440770807552736993877504, 12.08773311149594736791268653854, 13.37273763494693068873023434816, 14.77384342121426739201868467417

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