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Properties

Label 2-126-21.11-c8-0-4
Degree $2$
Conductor $126$
Sign $0.403 - 0.914i$
Analytic cond. $51.3297$
Root an. cond. $7.16447$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.79 − 5.65i)2-s + (63.9 − 110. i)4-s + (−846. + 488. i)5-s + (1.03e3 − 2.16e3i)7-s − 1.44e3i·8-s + (−5.52e3 + 9.57e3i)10-s + (−8.69e3 − 5.01e3i)11-s + 2.83e4·13-s + (−2.16e3 − 2.70e4i)14-s + (−8.19e3 − 1.41e4i)16-s + (−1.30e5 − 7.55e4i)17-s + (9.01e4 + 1.56e5i)19-s + 1.25e5i·20-s − 1.13e5·22-s + (1.79e5 − 1.03e5i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.35 + 0.781i)5-s + (0.429 − 0.903i)7-s − 0.353i·8-s + (−0.552 + 0.957i)10-s + (−0.593 − 0.342i)11-s + 0.993·13-s + (−0.0562 − 0.704i)14-s + (−0.125 − 0.216i)16-s + (−1.56 − 0.904i)17-s + (0.691 + 1.19i)19-s + 0.781i·20-s − 0.484·22-s + (0.639 − 0.369i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.403 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.403 - 0.914i$
Analytic conductor: \(51.3297\)
Root analytic conductor: \(7.16447\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :4),\ 0.403 - 0.914i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.429684531\)
\(L(\frac12)\) \(\approx\) \(1.429684531\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.79 + 5.65i)T \)
3 \( 1 \)
7 \( 1 + (-1.03e3 + 2.16e3i)T \)
good5 \( 1 + (846. - 488. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (8.69e3 + 5.01e3i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 2.83e4T + 8.15e8T^{2} \)
17 \( 1 + (1.30e5 + 7.55e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (-9.01e4 - 1.56e5i)T + (-8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-1.79e5 + 1.03e5i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 1.19e6iT - 5.00e11T^{2} \)
31 \( 1 + (8.82e5 - 1.52e6i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-1.01e6 - 1.75e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 - 3.62e5iT - 7.98e12T^{2} \)
43 \( 1 - 1.49e6T + 1.16e13T^{2} \)
47 \( 1 + (2.76e6 - 1.59e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-1.01e7 - 5.84e6i)T + (3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (6.93e6 + 4.00e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-9.78e6 - 1.69e7i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-2.39e6 + 4.14e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 2.83e7iT - 6.45e14T^{2} \)
73 \( 1 + (1.43e7 - 2.49e7i)T + (-4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (6.04e6 + 1.04e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 5.87e7iT - 2.25e15T^{2} \)
89 \( 1 + (8.51e6 - 4.91e6i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 - 8.66e7T + 7.83e15T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.80949141839176495334124131043, −10.95809036674641548411603573136, −10.59346635746770226482665692887, −8.690994365427688768421913299204, −7.49861959091606708869306139611, −6.70796244177894952413298798218, −4.98684136138007932030932947647, −3.85280340960953214050321089047, −3.02781516323766436402117937408, −1.13175378876187687980316029242, 0.34090971540220594020104451413, 2.25085074569144617215946504676, 3.88691463155006029605745557882, 4.74015188245983855758154722979, 5.90320342255189577116653508788, 7.40636837601741947265577035713, 8.320100976388463809562216258933, 9.096700465716732945882908538727, 11.26107864822826913518906440248, 11.50636162292896449130719103035

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