↑

Properties

Label 2-126-7.4-c9-0-1
Degree $2$
Conductor $126$
Sign $0.0658 - 0.997i$
Analytic cond. $64.8945$
Root an. cond. $8.05571$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−8 − 13.8i)2-s + (−127. + 221. i)4-s + (−84.1 − 145. i)5-s + (−3.16e3 − 5.50e3i)7-s + 4.09e3·8-s + (−1.34e3 + 2.33e3i)10-s + (7.24e3 − 1.25e4i)11-s − 1.09e5·13-s + (−5.10e4 + 8.78e4i)14-s + (−3.27e4 − 5.67e4i)16-s + (1.50e5 − 2.60e5i)17-s + (2.45e5 + 4.25e5i)19-s + 4.30e4·20-s − 2.31e5·22-s + (−1.13e6 − 1.96e6i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.0602 − 0.104i)5-s + (−0.497 − 0.867i)7-s + 0.353·8-s + (−0.0425 + 0.0737i)10-s + (0.149 − 0.258i)11-s − 1.05·13-s + (−0.355 + 0.611i)14-s + (−0.125 − 0.216i)16-s + (0.436 − 0.755i)17-s + (0.432 + 0.749i)19-s + 0.0602·20-s − 0.211·22-s + (−0.845 − 1.46i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0658 - 0.997i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0658 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.0658 - 0.997i$
Analytic conductor: \(64.8945\)
Root analytic conductor: \(8.05571\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :9/2),\ 0.0658 - 0.997i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.09032705515\)
\(L(\frac12)\) \(\approx\) \(0.09032705515\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (8 + 13.8i)T \)
3 \( 1 \)
7 \( 1 + (3.16e3 + 5.50e3i)T \)
good5 \( 1 + (84.1 + 145. i)T + (-9.76e5 + 1.69e6i)T^{2} \)
11 \( 1 + (-7.24e3 + 1.25e4i)T + (-1.17e9 - 2.04e9i)T^{2} \)
13 \( 1 + 1.09e5T + 1.06e10T^{2} \)
17 \( 1 + (-1.50e5 + 2.60e5i)T + (-5.92e10 - 1.02e11i)T^{2} \)
19 \( 1 + (-2.45e5 - 4.25e5i)T + (-1.61e11 + 2.79e11i)T^{2} \)
23 \( 1 + (1.13e6 + 1.96e6i)T + (-9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 - 3.80e6T + 1.45e13T^{2} \)
31 \( 1 + (3.95e6 - 6.84e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (7.07e6 + 1.22e7i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 + 1.11e7T + 3.27e14T^{2} \)
43 \( 1 - 8.50e6T + 5.02e14T^{2} \)
47 \( 1 + (2.18e7 + 3.78e7i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + (2.75e7 - 4.77e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 + (2.45e7 - 4.24e7i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (-7.09e7 - 1.22e8i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (-4.93e7 + 8.55e7i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + 2.96e8T + 4.58e16T^{2} \)
73 \( 1 + (2.30e7 - 3.98e7i)T + (-2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (-2.76e8 - 4.78e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 - 5.43e8T + 1.86e17T^{2} \)
89 \( 1 + (-2.54e8 - 4.40e8i)T + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 + 1.18e9T + 7.60e17T^{2} \)
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

−12.03546459406096316164602795073, −10.53300093527177724648531087666, −10.05999965125704965155145260535, −8.853940991283554618001492208680, −7.67958953617587659913069165549, −6.65057007383376825321648133366, −4.95561279675679261228459250392, −3.74091209834951488230750612234, −2.54814142923988436315365942212, −0.975087492190289130404358726401, 0.02943523185028159663653747819, 1.76690780674848964033978427034, 3.24269450629463210522821012921, 4.92202032246724826372680983039, 5.94865330267514870558103583356, 7.08331098380973003138542672475, 8.101547936048280457291753314871, 9.352885586722034709321026207037, 9.923833508297517230099236795202, 11.41426485322492423992882202269

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