↑

Properties

Label 2-126-63.59-c1-0-6
Degree $2$
Conductor $126$
Sign $0.998 - 0.0614i$
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  + (0.866 + 0.5i)2-s + (1.33 − 1.10i)3-s + (0.499 + 0.866i)4-s − 0.0676·5-s + (1.70 − 0.290i)6-s + (−2.64 − 0.142i)7-s + 0.999i·8-s + (0.554 − 2.94i)9-s + (−0.0585 − 0.0338i)10-s + 3.92i·11-s + (1.62 + 0.601i)12-s + (−3.32 − 1.92i)13-s + (−2.21 − 1.44i)14-s + (−0.0901 + 0.0747i)15-s + (−0.5 + 0.866i)16-s + (−0.775 + 1.34i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.769 − 0.638i)3-s + (0.249 + 0.433i)4-s − 0.0302·5-s + (0.697 − 0.118i)6-s + (−0.998 − 0.0538i)7-s + 0.353i·8-s + (0.184 − 0.982i)9-s + (−0.0185 − 0.0106i)10-s + 1.18i·11-s + (0.468 + 0.173i)12-s + (−0.922 − 0.532i)13-s + (−0.592 − 0.385i)14-s + (−0.0232 + 0.0193i)15-s + (−0.125 + 0.216i)16-s + (−0.188 + 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.998 - 0.0614i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.998 - 0.0614i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62212 + 0.0498648i\)
\(L(\frac12)\) \(\approx\) \(1.62212 + 0.0498648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.33 + 1.10i)T \)
7 \( 1 + (2.64 + 0.142i)T \)
good5 \( 1 + 0.0676T + 5T^{2} \)
11 \( 1 - 3.92iT - 11T^{2} \)
13 \( 1 + (3.32 + 1.92i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.775 - 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.06 + 2.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.52iT - 23T^{2} \)
29 \( 1 + (-1.20 + 0.697i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.09 + 0.632i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.35 + 7.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.17 + 8.96i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.735 - 1.27i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.77 - 3.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.28 - 3.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.70 - 8.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0705 - 0.0407i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.67 - 13.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.30iT - 71T^{2} \)
73 \( 1 + (-6.12 - 3.53i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.42 + 5.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.93 + 6.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.84 + 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.363 + 0.209i)T + (48.5 - 84.0i)T^{2} \)
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

−13.37488166867927556427512443918, −12.61434989911543050388321790058, −11.87560644656233526913220305200, −10.00119834464424741119668960648, −9.169888756367214879372981473989, −7.53765776883854433150168664088, −7.09637681585122306279630303517, −5.62129656693423464141389553000, −3.88547051166509815596824930518, −2.51876518567562769831935236525, 2.72007760832045348308996943883, 3.76811405925761164730552551958, 5.20098881044495917992902491602, 6.63264980534861965434590858140, 8.153382883014399345373900717321, 9.459832497861069679782121076571, 10.10296773443966673787356834988, 11.36691667030752309440393335170, 12.44370049202177466832404291919, 13.64221488170818931124717340428

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