↑

Properties

Label 2-126-63.4-c1-0-7
Degree $2$
Conductor $126$
Sign $-0.778 + 0.627i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−1.09 − 1.34i)3-s + (−0.499 − 0.866i)4-s − 3.18·5-s + (−1.71 + 0.272i)6-s + (0.710 − 2.54i)7-s − 0.999·8-s + (−0.619 + 2.93i)9-s + (−1.59 + 2.75i)10-s + 3.18·11-s + (−0.619 + 1.61i)12-s + (2.85 − 4.93i)13-s + (−1.85 − 1.88i)14-s + (3.47 + 4.28i)15-s + (−0.5 + 0.866i)16-s + (−0.760 + 1.31i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.629 − 0.776i)3-s + (−0.249 − 0.433i)4-s − 1.42·5-s + (−0.698 + 0.111i)6-s + (0.268 − 0.963i)7-s − 0.353·8-s + (−0.206 + 0.978i)9-s + (−0.503 + 0.871i)10-s + 0.959·11-s + (−0.178 + 0.466i)12-s + (0.790 − 1.36i)13-s + (−0.494 − 0.505i)14-s + (0.896 + 1.10i)15-s + (−0.125 + 0.216i)16-s + (−0.184 + 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.778 + 0.627i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ -0.778 + 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273215 - 0.774285i\)
\(L(\frac12)\) \(\approx\) \(0.273215 - 0.774285i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (1.09 + 1.34i)T \)
7 \( 1 + (-0.710 + 2.54i)T \)
good5 \( 1 + 3.18T + 5T^{2} \)
11 \( 1 - 3.18T + 11T^{2} \)
13 \( 1 + (-2.85 + 4.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.760 - 1.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.23T + 23T^{2} \)
29 \( 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.71 - 8.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.02 + 1.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.562 - 0.974i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.56 - 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.48 + 9.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 + (2.48 - 4.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.06 + 3.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.03 + 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.112 - 0.195i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.42 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.84749584096421214894120455404, −11.84862307226971439691431353380, −11.19259231898487922115964248667, −10.41536900339418581714976708502, −8.441813217823046133040286222695, −7.53950574370617436301038422251, −6.33901154360039921833016422647, −4.69393714370206280513870960388, −3.51011320438515015745342506711, −0.933624983337636158523858647482, 3.72482347751956533205239016958, 4.54029461108714216190735535763, 5.95721394063676308002915931262, 7.07452238466565773544418706421, 8.598386425988124166028155274702, 9.252161766511034343310883989937, 11.18097526044979165918806812686, 11.66859255649554420025152577541, 12.42750116881381981572973108615, 14.13634867592788218311233677310

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