Properties

Label 2-43-43.9-c9-0-28
Degree $2$
Conductor $43$
Sign $0.309 + 0.951i$
Analytic cond. $22.1465$
Root an. cond. $4.70601$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (6.85 + 8.58i)2-s + (188. + 28.3i)3-s + (87.0 − 381. i)4-s + (−1.24e3 − 849. i)5-s + (1.04e3 + 1.81e3i)6-s + (−1.10e3 + 1.90e3i)7-s + (8.94e3 − 4.30e3i)8-s + (1.58e4 + 4.87e3i)9-s + (−1.23e3 − 1.65e4i)10-s + (−1.08e4 − 4.74e4i)11-s + (2.72e4 − 6.93e4i)12-s + (4.97e3 − 6.64e4i)13-s + (−2.39e4 + 3.60e3i)14-s + (−2.10e5 − 1.95e5i)15-s + (−8.22e4 − 3.96e4i)16-s + (1.29e5 − 8.84e4i)17-s + ⋯
L(s)  = 1  + (0.302 + 0.379i)2-s + (1.34 + 0.202i)3-s + (0.170 − 0.745i)4-s + (−0.891 − 0.607i)5-s + (0.329 + 0.570i)6-s + (−0.173 + 0.300i)7-s + (0.771 − 0.371i)8-s + (0.803 + 0.247i)9-s + (−0.0391 − 0.522i)10-s + (−0.223 − 0.977i)11-s + (0.378 − 0.965i)12-s + (0.0483 − 0.644i)13-s + (−0.166 + 0.0251i)14-s + (−1.07 − 0.995i)15-s + (−0.313 − 0.151i)16-s + (0.376 − 0.256i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.309 + 0.951i$
Analytic conductor: \(22.1465\)
Root analytic conductor: \(4.70601\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :9/2),\ 0.309 + 0.951i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.30356 - 1.67339i\)
\(L(\frac12)\) \(\approx\) \(2.30356 - 1.67339i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-2.07e7 + 8.38e6i)T \)
good2 \( 1 + (-6.85 - 8.58i)T + (-113. + 499. i)T^{2} \)
3 \( 1 + (-188. - 28.3i)T + (1.88e4 + 5.80e3i)T^{2} \)
5 \( 1 + (1.24e3 + 849. i)T + (7.13e5 + 1.81e6i)T^{2} \)
7 \( 1 + (1.10e3 - 1.90e3i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (1.08e4 + 4.74e4i)T + (-2.12e9 + 1.02e9i)T^{2} \)
13 \( 1 + (-4.97e3 + 6.64e4i)T + (-1.04e10 - 1.58e9i)T^{2} \)
17 \( 1 + (-1.29e5 + 8.84e4i)T + (4.33e10 - 1.10e11i)T^{2} \)
19 \( 1 + (5.74e4 - 1.77e4i)T + (2.66e11 - 1.81e11i)T^{2} \)
23 \( 1 + (-4.04e5 + 3.75e5i)T + (1.34e11 - 1.79e12i)T^{2} \)
29 \( 1 + (1.73e6 - 2.61e5i)T + (1.38e13 - 4.27e12i)T^{2} \)
31 \( 1 + (-2.12e6 + 5.40e6i)T + (-1.93e13 - 1.79e13i)T^{2} \)
37 \( 1 + (-7.31e6 - 1.26e7i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 + (-1.46e7 - 1.83e7i)T + (-7.28e13 + 3.19e14i)T^{2} \)
47 \( 1 + (-1.68e6 + 7.40e6i)T + (-1.00e15 - 4.85e14i)T^{2} \)
53 \( 1 + (1.38e6 + 1.85e7i)T + (-3.26e15 + 4.91e14i)T^{2} \)
59 \( 1 + (-4.63e7 - 2.23e7i)T + (5.40e15 + 6.77e15i)T^{2} \)
61 \( 1 + (-3.45e6 - 8.79e6i)T + (-8.57e15 + 7.95e15i)T^{2} \)
67 \( 1 + (1.81e8 - 5.60e7i)T + (2.24e16 - 1.53e16i)T^{2} \)
71 \( 1 + (-3.72e7 - 3.45e7i)T + (3.42e15 + 4.57e16i)T^{2} \)
73 \( 1 + (2.12e7 - 2.83e8i)T + (-5.82e16 - 8.77e15i)T^{2} \)
79 \( 1 + (2.80e8 - 4.86e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + (-3.37e8 - 5.08e7i)T + (1.78e17 + 5.51e16i)T^{2} \)
89 \( 1 + (-5.78e7 - 8.71e6i)T + (3.34e17 + 1.03e17i)T^{2} \)
97 \( 1 + (3.32e8 + 1.45e9i)T + (-6.84e17 + 3.29e17i)T^{2} \)
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

−13.96284263850524907125804236717, −12.95899768307801584653228752215, −11.32831719051861655594818071980, −9.827688373215205902467883947866, −8.595151330942857346261955315990, −7.68230158753875923598195067261, −5.80826668620662193928427048218, −4.24549049765333102487608964537, −2.79390148971936040485532862585, −0.76438461265527539329133652121, 2.08442433570703936742753431241, 3.26210843502641428156742535447, 4.18139049352196155369462086789, 7.20593269439838185291150496821, 7.74391029853059711043773586836, 9.101017483032494626029606731222, 10.76494059389269286910442775633, 12.03461149726232278286874734329, 13.06258587780196904133272713765, 14.16236364534006930176751411231

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