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Properties

Label 2-126-21.11-c6-0-12
Degree $2$
Conductor $126$
Sign $-0.248 + 0.968i$
Analytic cond. $28.9868$
Root an. cond. $5.38394$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.89 − 2.82i)2-s + (15.9 − 27.7i)4-s + (−20.9 + 12.1i)5-s + (94.7 − 329. i)7-s − 181. i·8-s + (−68.4 + 118. i)10-s + (1.97e3 + 1.13e3i)11-s + 482.·13-s + (−467. − 1.88e3i)14-s + (−512. − 886. i)16-s + (−7.77e3 − 4.49e3i)17-s + (−3.18e3 − 5.52e3i)19-s + 774. i·20-s + 1.28e4·22-s + (1.26e4 − 7.29e3i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.167 + 0.0968i)5-s + (0.276 − 0.961i)7-s − 0.353i·8-s + (−0.0684 + 0.118i)10-s + (1.48 + 0.855i)11-s + 0.219·13-s + (−0.170 − 0.686i)14-s + (−0.125 − 0.216i)16-s + (−1.58 − 0.914i)17-s + (−0.465 − 0.805i)19-s + 0.0968i·20-s + 1.21·22-s + (1.03 − 0.599i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.248 + 0.968i$
Analytic conductor: \(28.9868\)
Root analytic conductor: \(5.38394\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3),\ -0.248 + 0.968i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.662274645\)
\(L(\frac12)\) \(\approx\) \(2.662274645\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.89 + 2.82i)T \)
3 \( 1 \)
7 \( 1 + (-94.7 + 329. i)T \)
good5 \( 1 + (20.9 - 12.1i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-1.97e3 - 1.13e3i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 482.T + 4.82e6T^{2} \)
17 \( 1 + (7.77e3 + 4.49e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (3.18e3 + 5.52e3i)T + (-2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-1.26e4 + 7.29e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 3.40e4iT - 5.94e8T^{2} \)
31 \( 1 + (-2.60e4 + 4.51e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-986. - 1.70e3i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 + 6.54e3iT - 4.75e9T^{2} \)
43 \( 1 + 1.36e5T + 6.32e9T^{2} \)
47 \( 1 + (-1.57e5 + 9.08e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-5.28e4 - 3.05e4i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-2.03e5 - 1.17e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.57e5 + 2.73e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.43e5 - 2.47e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 2.11e5iT - 1.28e11T^{2} \)
73 \( 1 + (1.36e5 - 2.36e5i)T + (-7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-1.33e5 - 2.31e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 1.86e5iT - 3.26e11T^{2} \)
89 \( 1 + (3.28e5 - 1.89e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.01e6T + 8.32e11T^{2} \)
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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

−11.68091580544913207157761896960, −11.26706977180151838436646596958, −9.967686089121208173547498100460, −8.919891525976216081070366654321, −7.21817715390112123258349901004, −6.52174773070039651957865076415, −4.65365182894405917600394813930, −3.99640902401207293949391668709, −2.23855711774423489496595312139, −0.70464454781078130609772977280, 1.62908175476449154018428615567, 3.33433735540042457524803778596, 4.56495294888403179774082922376, 5.95319860759786102388218924283, 6.73278616973869905730379212373, 8.491463828563557625407666092743, 8.894051466132610242033348421575, 10.75830486146283813288444217509, 11.71615396441437774513027716319, 12.48143079012876220712466546412

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