Properties

Label 2-126-7.6-c2-0-4
Degree $2$
Conductor $126$
Sign $0.320 + 0.947i$
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  − 1.41·2-s + 2.00·4-s − 1.01i·5-s + (−2.24 − 6.63i)7-s − 2.82·8-s + 1.43i·10-s + 10.2·11-s − 8.95i·13-s + (3.17 + 9.37i)14-s + 4.00·16-s − 30.4i·17-s − 16.1i·19-s − 2.02i·20-s − 14.4·22-s + 6.72·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.500·4-s − 0.202i·5-s + (−0.320 − 0.947i)7-s − 0.353·8-s + 0.143i·10-s + 0.931·11-s − 0.689i·13-s + (0.226 + 0.669i)14-s + 0.250·16-s − 1.78i·17-s − 0.849i·19-s − 0.101i·20-s − 0.658·22-s + 0.292·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.320 + 0.947i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.320 + 0.947i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.762130 - 0.546782i\)
\(L(\frac12)\) \(\approx\) \(0.762130 - 0.546782i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 \)
7 \( 1 + (2.24 + 6.63i)T \)
good5 \( 1 + 1.01iT - 25T^{2} \)
11 \( 1 - 10.2T + 121T^{2} \)
13 \( 1 + 8.95iT - 169T^{2} \)
17 \( 1 + 30.4iT - 289T^{2} \)
19 \( 1 + 16.1iT - 361T^{2} \)
23 \( 1 - 6.72T + 529T^{2} \)
29 \( 1 + 30T + 841T^{2} \)
31 \( 1 - 50.1iT - 961T^{2} \)
37 \( 1 - 30.9T + 1.36e3T^{2} \)
41 \( 1 - 7.10iT - 1.68e3T^{2} \)
43 \( 1 + 74.4T + 1.84e3T^{2} \)
47 \( 1 - 58.2iT - 2.20e3T^{2} \)
53 \( 1 - 70.9T + 2.80e3T^{2} \)
59 \( 1 - 0.492iT - 3.48e3T^{2} \)
61 \( 1 - 2.86iT - 3.72e3T^{2} \)
67 \( 1 - 27.0T + 4.48e3T^{2} \)
71 \( 1 + 50.6T + 5.04e3T^{2} \)
73 \( 1 + 70.6iT - 5.32e3T^{2} \)
79 \( 1 - 133.T + 6.24e3T^{2} \)
83 \( 1 - 104. iT - 6.88e3T^{2} \)
89 \( 1 + 144. iT - 7.92e3T^{2} \)
97 \( 1 + 100. iT - 9.40e3T^{2} \)
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

−12.92225666444986204158210886226, −11.69240753940359883923042948398, −10.78601630673230357617528566374, −9.677626717277536076238031986941, −8.859971293110088272252780814349, −7.43083754234991012902611636781, −6.67375099281273375981940689910, −4.90243776562815311854785785491, −3.16177521355316739651267637864, −0.862224439222901504643703981763, 1.92308933130853592037675298565, 3.77263563162550108011292070486, 5.83784777941426809633355268342, 6.73887302093743430002416618475, 8.229454515596870104978422256300, 9.095807131029857804267062996500, 10.05773891608081051573281535019, 11.28189637957805225355009071598, 12.12228356461809682312169207122, 13.15533207347935499115924148928

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