↑

Properties

Label 2-126-21.20-c3-0-6
Degree $2$
Conductor $126$
Sign $-0.0928 + 0.995i$
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  + 2i·2-s − 4·4-s − 14.3·5-s + (9.24 + 16.0i)7-s − 8i·8-s − 28.6i·10-s − 59.6i·11-s − 53.7i·13-s + (−32.0 + 18.4i)14-s + 16·16-s − 135.·17-s − 82.4i·19-s + 57.2·20-s + 119.·22-s + 20.7i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.28·5-s + (0.499 + 0.866i)7-s − 0.353i·8-s − 0.905i·10-s − 1.63i·11-s − 1.14i·13-s + (−0.612 + 0.352i)14-s + 0.250·16-s − 1.93·17-s − 0.994i·19-s + 0.640·20-s + 1.15·22-s + 0.188i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0928 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0928 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.0928 + 0.995i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ -0.0928 + 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.273393 - 0.300070i\)
\(L(\frac12)\) \(\approx\) \(0.273393 - 0.300070i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 \)
7 \( 1 + (-9.24 - 16.0i)T \)
good5 \( 1 + 14.3T + 125T^{2} \)
11 \( 1 + 59.6iT - 1.33e3T^{2} \)
13 \( 1 + 53.7iT - 2.19e3T^{2} \)
17 \( 1 + 135.T + 4.91e3T^{2} \)
19 \( 1 + 82.4iT - 6.85e3T^{2} \)
23 \( 1 - 20.7iT - 1.21e4T^{2} \)
29 \( 1 - 63.0iT - 2.43e4T^{2} \)
31 \( 1 - 121. iT - 2.97e4T^{2} \)
37 \( 1 + 325.T + 5.06e4T^{2} \)
41 \( 1 - 206.T + 6.89e4T^{2} \)
43 \( 1 - 274.T + 7.95e4T^{2} \)
47 \( 1 + 294.T + 1.03e5T^{2} \)
53 \( 1 + 178. iT - 1.48e5T^{2} \)
59 \( 1 - 170.T + 2.05e5T^{2} \)
61 \( 1 - 72.8iT - 2.26e5T^{2} \)
67 \( 1 + 1.05e3T + 3.00e5T^{2} \)
71 \( 1 - 28.8iT - 3.57e5T^{2} \)
73 \( 1 + 482. iT - 3.89e5T^{2} \)
79 \( 1 + 707.T + 4.93e5T^{2} \)
83 \( 1 + 778.T + 5.71e5T^{2} \)
89 \( 1 - 9.89T + 7.04e5T^{2} \)
97 \( 1 + 1.56e3iT - 9.12e5T^{2} \)
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

−12.71238303248018429271386416349, −11.47742530206066587001834803220, −10.86734282913785799814351521553, −8.780240053106218703039027599246, −8.488299910734887712511586345649, −7.26003311630417688391530220597, −5.89685486550434341936524451179, −4.68096032745289345300077091307, −3.15577559434180688972368488978, −0.20800433207986027113895329119, 1.91800555145246860187022257155, 4.13623193190279228757274469127, 4.44617951493962336392761792845, 6.89548649822072562312891369693, 7.78238724981405181933493045324, 9.029042601205469794371875000196, 10.26570222848677182642781032783, 11.27666698022077671351552881849, 11.94502166215011332108636926916, 12.95024145176505572731390582558

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