↑

Properties

Label 2-126-63.41-c3-0-8
Degree $2$
Conductor $126$
Sign $0.999 + 0.00353i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − i)2-s + (−5.19 + 0.0586i)3-s + (1.99 − 3.46i)4-s + (−2.95 + 5.12i)5-s + (−8.94 + 5.29i)6-s + (13.8 + 12.2i)7-s − 7.99i·8-s + (26.9 − 0.609i)9-s + 11.8i·10-s + (28.3 − 16.3i)11-s + (−10.1 + 18.1i)12-s + (35.4 + 20.4i)13-s + (36.2 + 7.38i)14-s + (15.0 − 26.7i)15-s + (−8 − 13.8i)16-s + 58.5·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.999 + 0.0112i)3-s + (0.249 − 0.433i)4-s + (−0.264 + 0.458i)5-s + (−0.608 + 0.360i)6-s + (0.748 + 0.662i)7-s − 0.353i·8-s + (0.999 − 0.0225i)9-s + 0.374i·10-s + (0.778 − 0.449i)11-s + (−0.245 + 0.435i)12-s + (0.757 + 0.437i)13-s + (0.692 + 0.140i)14-s + (0.259 − 0.461i)15-s + (−0.125 − 0.216i)16-s + 0.835·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00353i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.999 + 0.00353i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.999 + 0.00353i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.85292 - 0.00327066i\)
\(L(\frac12)\) \(\approx\) \(1.85292 - 0.00327066i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + i)T \)
3 \( 1 + (5.19 - 0.0586i)T \)
7 \( 1 + (-13.8 - 12.2i)T \)
good5 \( 1 + (2.95 - 5.12i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-28.3 + 16.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-35.4 - 20.4i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 58.5T + 4.91e3T^{2} \)
19 \( 1 + 11.3iT - 6.85e3T^{2} \)
23 \( 1 + (-21.4 - 12.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (62.1 - 35.8i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-175. - 101. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 208.T + 5.06e4T^{2} \)
41 \( 1 + (3.41 - 5.91i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-16.6 - 28.7i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (284. + 492. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 180. iT - 1.48e5T^{2} \)
59 \( 1 + (-282. + 488. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-247. + 142. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (260. - 451. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 557. iT - 3.57e5T^{2} \)
73 \( 1 - 978. iT - 3.89e5T^{2} \)
79 \( 1 + (665. + 1.15e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (323. + 560. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 646.T + 7.04e5T^{2} \)
97 \( 1 + (382. - 221. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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

−12.64750620123313198829657166521, −11.59878027963617018501544555116, −11.33544806889570151684381174796, −10.16552808569294654689670629804, −8.710489640157287918988120634644, −7.07942154997598119930691236455, −6.00623994921215458719453441862, −4.96038659759841983502939678191, −3.54962943577893227460402658147, −1.42651683399316990795032696716, 1.15699761157562640849071980131, 3.96002714963696670020069126494, 4.88418185987936912127814804959, 6.08822334816311432985691287445, 7.24997221366699171032087453905, 8.325449984692430930356977176711, 10.01731920950249873827947852141, 11.12961514856053566295788599907, 11.95088874289257113325490739172, 12.78335264019875821509483331297

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