↑

Properties

Label 2-126-63.2-c2-0-10
Degree $2$
Conductor $126$
Sign $0.854 + 0.519i$
Analytic cond. $3.43325$
Root an. cond. $1.85290$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.953 + 2.84i)3-s + (0.999 − 1.73i)4-s − 9.69i·5-s + (3.17 + 2.80i)6-s + (6.99 + 0.244i)7-s − 2.82i·8-s + (−7.17 + 5.42i)9-s + (−6.85 − 11.8i)10-s − 0.747i·11-s + (5.88 + 1.19i)12-s + (8.35 + 14.4i)13-s + (8.74 − 4.64i)14-s + (27.5 − 9.25i)15-s + (−2.00 − 3.46i)16-s + (−5.71 + 3.30i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.317 + 0.948i)3-s + (0.249 − 0.433i)4-s − 1.93i·5-s + (0.529 + 0.468i)6-s + (0.999 + 0.0349i)7-s − 0.353i·8-s + (−0.797 + 0.602i)9-s + (−0.685 − 1.18i)10-s − 0.0679i·11-s + (0.490 + 0.0993i)12-s + (0.642 + 1.11i)13-s + (0.624 − 0.331i)14-s + (1.83 − 0.616i)15-s + (−0.125 − 0.216i)16-s + (−0.336 + 0.194i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.854 + 0.519i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.854 + 0.519i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.12467 - 0.595045i\)
\(L(\frac12)\) \(\approx\) \(2.12467 - 0.595045i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (-0.953 - 2.84i)T \)
7 \( 1 + (-6.99 - 0.244i)T \)
good5 \( 1 + 9.69iT - 25T^{2} \)
11 \( 1 + 0.747iT - 121T^{2} \)
13 \( 1 + (-8.35 - 14.4i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (5.71 - 3.30i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (0.429 - 0.744i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 - 19.1iT - 529T^{2} \)
29 \( 1 + (19.2 + 11.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-19.6 + 34.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (34.2 - 59.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-27.4 + 15.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (30.3 - 52.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-1.03 + 0.597i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-20.6 + 11.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-19.6 - 11.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-4.17 - 7.22i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-8.09 + 14.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 36.6iT - 5.04e3T^{2} \)
73 \( 1 + (-1.75 - 3.03i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (50.1 + 86.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-21.5 - 12.4i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (53.5 + 30.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-22.1 + 38.4i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−13.26340594130521584581239967062, −11.83383415525040090381656974249, −11.32931514056767560984777158207, −9.787756453685641252521936943586, −8.894077146220320846202881160561, −8.096552569541809501768186864244, −5.72440524926124013242085843100, −4.72225693560022098168281848483, −4.06233524019860060251401275909, −1.68057414581444924967381925094, 2.33882474524093218994056476557, 3.52441463537087681052410837297, 5.64946288205824715145417962220, 6.76916915293005654779746620617, 7.49183220909284376103936838470, 8.501198978237017277199449566950, 10.56305294154332258108112484109, 11.20240690525479162965319807224, 12.30658346990538695819437539383, 13.54124311291891890495408720736

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