↑

Properties

Label 2-126-63.11-c2-0-15
Degree $2$
Conductor $126$
Sign $-0.751 - 0.659i$
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  − 1.41i·2-s + (−0.991 − 2.83i)3-s − 2.00·4-s + (−6.36 − 3.67i)5-s + (−4.00 + 1.40i)6-s + (2.82 + 6.40i)7-s + 2.82i·8-s + (−7.03 + 5.61i)9-s + (−5.19 + 9.00i)10-s + (3.62 − 2.09i)11-s + (1.98 + 5.66i)12-s + (−0.596 − 1.03i)13-s + (9.05 − 3.99i)14-s + (−4.09 + 21.6i)15-s + 4.00·16-s + (−17.8 − 10.2i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.330 − 0.943i)3-s − 0.500·4-s + (−1.27 − 0.735i)5-s + (−0.667 + 0.233i)6-s + (0.403 + 0.915i)7-s + 0.353i·8-s + (−0.781 + 0.623i)9-s + (−0.519 + 0.900i)10-s + (0.329 − 0.190i)11-s + (0.165 + 0.471i)12-s + (−0.0458 − 0.0794i)13-s + (0.647 − 0.285i)14-s + (−0.273 + 1.44i)15-s + 0.250·16-s + (−1.04 − 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.751 - 0.659i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.751 - 0.659i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.161649 + 0.429424i\)
\(L(\frac12)\) \(\approx\) \(0.161649 + 0.429424i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (0.991 + 2.83i)T \)
7 \( 1 + (-2.82 - 6.40i)T \)
good5 \( 1 + (6.36 + 3.67i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.62 + 2.09i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (0.596 + 1.03i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (17.8 + 10.2i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (9.10 + 15.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (21.6 + 12.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (2.85 + 1.65i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 51.9T + 961T^{2} \)
37 \( 1 + (-9.43 - 16.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-66.2 + 38.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-25.6 + 44.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 59.3iT - 2.20e3T^{2} \)
53 \( 1 + (-1.33 - 0.767i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 32.0iT - 3.48e3T^{2} \)
61 \( 1 - 111.T + 3.72e3T^{2} \)
67 \( 1 + 28.1T + 4.48e3T^{2} \)
71 \( 1 - 75.9iT - 5.04e3T^{2} \)
73 \( 1 + (-31.0 + 53.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 67.1T + 6.24e3T^{2} \)
83 \( 1 + (64.7 + 37.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (95.3 - 55.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-14.0 + 24.3i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−12.35483105322166799588920692718, −11.62206398523102667302974934563, −11.01853080339670707323174449060, −8.971234013632683242003486477176, −8.409860514055597394621323743248, −7.17926603630295652811731241114, −5.52310764867213645424839126558, −4.24935210823206506237933463125, −2.27996738442029975546372173229, −0.32520906326826546632077365180, 3.80712771507195142606983146986, 4.34171496130651401214029659619, 6.09386184535603922127087471974, 7.31653042599476855496270502803, 8.235250302698070427017813541863, 9.612411376175592730984488153034, 10.84782038367860617268896207095, 11.32853473879961715018830550202, 12.72397210273656959513132834249, 14.43159052618940094123985739780

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