Properties

Degree 2
Conductor 43
Sign $-0.0401 - 0.999i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 7.11i·2-s + (−393. + 227. i)3-s + 973.·4-s + (−1.94e3 + 1.12e3i)5-s + (−1.61e3 − 2.79e3i)6-s + (8.90e3 + 5.14e3i)7-s + 1.42e4i·8-s + (7.37e4 − 1.27e5i)9-s + (−7.97e3 − 1.38e4i)10-s + 2.29e5·11-s + (−3.83e5 + 2.21e5i)12-s + (5.23e4 − 9.06e4i)13-s + (−3.65e4 + 6.33e4i)14-s + (5.10e5 − 8.83e5i)15-s + 8.95e5·16-s + (9.78e5 − 1.69e6i)17-s + ⋯
L(s)  = 1  + 0.222i·2-s + (−1.61 + 0.935i)3-s + 0.950·4-s + (−0.621 + 0.359i)5-s + (−0.207 − 0.359i)6-s + (0.530 + 0.306i)7-s + 0.433i·8-s + (1.24 − 2.16i)9-s + (−0.0797 − 0.138i)10-s + 1.42·11-s + (−1.53 + 0.889i)12-s + (0.140 − 0.244i)13-s + (−0.0680 + 0.117i)14-s + (0.671 − 1.16i)15-s + 0.854·16-s + (0.689 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0401 - 0.999i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.0401 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.0401 - 0.999i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.0401 - 0.999i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(1.06127 + 1.10480i\)
\(L(\frac12)\)  \(\approx\)  \(1.06127 + 1.10480i\)
\(L(6)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 + (-1.46e8 - 6.76e6i)T \)
good2 \( 1 - 7.11iT - 1.02e3T^{2} \)
3 \( 1 + (393. - 227. i)T + (2.95e4 - 5.11e4i)T^{2} \)
5 \( 1 + (1.94e3 - 1.12e3i)T + (4.88e6 - 8.45e6i)T^{2} \)
7 \( 1 + (-8.90e3 - 5.14e3i)T + (1.41e8 + 2.44e8i)T^{2} \)
11 \( 1 - 2.29e5T + 2.59e10T^{2} \)
13 \( 1 + (-5.23e4 + 9.06e4i)T + (-6.89e10 - 1.19e11i)T^{2} \)
17 \( 1 + (-9.78e5 + 1.69e6i)T + (-1.00e12 - 1.74e12i)T^{2} \)
19 \( 1 + (-9.72e4 + 5.61e4i)T + (3.06e12 - 5.30e12i)T^{2} \)
23 \( 1 + (-2.53e6 - 4.38e6i)T + (-2.07e13 + 3.58e13i)T^{2} \)
29 \( 1 + (-1.83e6 - 1.05e6i)T + (2.10e14 + 3.64e14i)T^{2} \)
31 \( 1 + (-3.80e6 - 6.59e6i)T + (-4.09e14 + 7.09e14i)T^{2} \)
37 \( 1 + (-2.72e7 + 1.57e7i)T + (2.40e15 - 4.16e15i)T^{2} \)
41 \( 1 - 2.35e7T + 1.34e16T^{2} \)
47 \( 1 + 2.97e8T + 5.25e16T^{2} \)
53 \( 1 + (-2.77e8 - 4.80e8i)T + (-8.74e16 + 1.51e17i)T^{2} \)
59 \( 1 - 1.27e9T + 5.11e17T^{2} \)
61 \( 1 + (1.25e9 + 7.22e8i)T + (3.56e17 + 6.17e17i)T^{2} \)
67 \( 1 + (6.63e8 + 1.14e9i)T + (-9.11e17 + 1.57e18i)T^{2} \)
71 \( 1 + (-2.42e8 - 1.39e8i)T + (1.62e18 + 2.81e18i)T^{2} \)
73 \( 1 + (6.34e8 + 3.66e8i)T + (2.14e18 + 3.72e18i)T^{2} \)
79 \( 1 + (-1.18e9 + 2.05e9i)T + (-4.73e18 - 8.19e18i)T^{2} \)
83 \( 1 + (-3.79e9 - 6.57e9i)T + (-7.75e18 + 1.34e19i)T^{2} \)
89 \( 1 + (2.16e9 - 1.25e9i)T + (1.55e19 - 2.70e19i)T^{2} \)
97 \( 1 - 1.27e10T + 7.37e19T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.62294503416072735570482840595, −12.05640119803561387853189281204, −11.59603643558507724922980072861, −10.87777799185080652441250968335, −9.485757070416482520915912141616, −7.35477935896566686520528413518, −6.24098415993517735979667941177, −5.10218443235544882766531758186, −3.57139909767217306022993449851, −1.05489223419545132244472693191, 0.825533322160035317192832135487, 1.64478642549891102970730379447, 4.24523276624264492634677652551, 5.96235487023972731859382560263, 6.84731452199121340759651925335, 7.997335761667299110528416905758, 10.42217640516295987979116403982, 11.46000001347447749633515901624, 11.94692963049914946840935134209, 12.85254596014281252980429041941

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