↑

Properties

Label 2-126-21.17-c11-0-22
Degree $2$
Conductor $126$
Sign $-0.157 + 0.987i$
Analytic cond. $96.8112$
Root an. cond. $9.83927$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−27.7 + 16i)2-s + (511. − 886. i)4-s + (−419. − 727. i)5-s + (3.48e4 − 2.76e4i)7-s + 3.27e4i·8-s + (2.32e4 + 1.34e4i)10-s + (2.04e5 + 1.18e5i)11-s − 1.99e6i·13-s + (−5.23e5 + 1.32e6i)14-s + (−5.24e5 − 9.08e5i)16-s + (2.36e6 − 4.10e6i)17-s + (−3.03e6 + 1.75e6i)19-s − 8.59e5·20-s − 7.55e6·22-s + (1.80e7 − 1.04e7i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0600 − 0.104i)5-s + (0.783 − 0.621i)7-s + 0.353i·8-s + (0.0735 + 0.0424i)10-s + (0.382 + 0.221i)11-s − 1.48i·13-s + (−0.260 + 0.657i)14-s + (−0.125 − 0.216i)16-s + (0.404 − 0.701i)17-s + (−0.281 + 0.162i)19-s − 0.0600·20-s − 0.312·22-s + (0.585 − 0.338i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.157 + 0.987i$
Analytic conductor: \(96.8112\)
Root analytic conductor: \(9.83927\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :11/2),\ -0.157 + 0.987i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.480303664\)
\(L(\frac12)\) \(\approx\) \(1.480303664\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (27.7 - 16i)T \)
3 \( 1 \)
7 \( 1 + (-3.48e4 + 2.76e4i)T \)
good5 \( 1 + (419. + 727. i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-2.04e5 - 1.18e5i)T + (1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 + 1.99e6iT - 1.79e12T^{2} \)
17 \( 1 + (-2.36e6 + 4.10e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (3.03e6 - 1.75e6i)T + (5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (-1.80e7 + 1.04e7i)T + (4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 + 9.71e6iT - 1.22e16T^{2} \)
31 \( 1 + (-4.04e7 - 2.33e7i)T + (1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (7.55e6 + 1.30e7i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 + 3.74e8T + 5.50e17T^{2} \)
43 \( 1 + 4.25e8T + 9.29e17T^{2} \)
47 \( 1 + (-9.51e8 - 1.64e9i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (7.33e7 + 4.23e7i)T + (4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (-1.90e9 + 3.30e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (2.55e9 - 1.47e9i)T + (2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (9.25e9 - 1.60e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 - 2.98e10iT - 2.31e20T^{2} \)
73 \( 1 + (1.66e10 + 9.60e9i)T + (1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (1.94e9 + 3.36e9i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 1.89e10T + 1.28e21T^{2} \)
89 \( 1 + (-4.21e9 - 7.29e9i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 + 4.89e10iT - 7.15e21T^{2} \)
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

−10.71761111757702345754662270687, −10.01044905475381927459104166916, −8.670114037119507063697339124922, −7.85033431021351846822324299133, −6.91311286113016133623378066047, −5.54346170651923185413361260134, −4.46065665070248811255629019255, −2.85553745916294180844101787599, −1.30198727808097819258928673855, −0.43934504901640559285584339521, 1.25235071264673510717113624684, 2.10448366704516752078688756954, 3.54912635118866772697130014847, 4.85260420434084785644097750982, 6.30029861866399766703868626060, 7.44410828145251739554918493745, 8.652590020322586981477234056535, 9.247551375869748187358851243731, 10.60402980112288482620564788835, 11.50738414899005814106010401818

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