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Properties

Label 2-126-21.11-c8-0-12
Degree $2$
Conductor $126$
Sign $-0.242 + 0.970i$
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 + (−731. + 422. i)5-s + (−647. + 2.31e3i)7-s − 1.44e3i·8-s + (−4.77e3 + 8.27e3i)10-s + (1.46e3 + 848. i)11-s − 1.20e4·13-s + (6.73e3 + 2.63e4i)14-s + (−8.19e3 − 1.41e4i)16-s + (4.83e4 + 2.79e4i)17-s + (−5.10e4 − 8.83e4i)19-s + 1.08e5i·20-s + 1.91e4·22-s + (1.33e5 − 7.71e4i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.17 + 0.675i)5-s + (−0.269 + 0.962i)7-s − 0.353i·8-s + (−0.477 + 0.827i)10-s + (0.100 + 0.0579i)11-s − 0.421·13-s + (0.175 + 0.685i)14-s + (−0.125 − 0.216i)16-s + (0.578 + 0.334i)17-s + (−0.391 − 0.678i)19-s + 0.675i·20-s + 0.0819·22-s + (0.477 − 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.242 + 0.970i$
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: \((2,\ 126,\ (\ :4),\ -0.242 + 0.970i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.224118015\)
\(L(\frac12)\) \(\approx\) \(1.224118015\)
\(L(5)\) 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 \( 1 + (647. - 2.31e3i)T \)
good5 \( 1 + (731. - 422. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-1.46e3 - 848. i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 1.20e4T + 8.15e8T^{2} \)
17 \( 1 + (-4.83e4 - 2.79e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (5.10e4 + 8.83e4i)T + (-8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-1.33e5 + 7.71e4i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 1.12e6iT - 5.00e11T^{2} \)
31 \( 1 + (-2.65e5 + 4.59e5i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (1.65e5 + 2.86e5i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 1.44e6iT - 7.98e12T^{2} \)
43 \( 1 + 9.41e5T + 1.16e13T^{2} \)
47 \( 1 + (3.11e5 - 1.79e5i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-3.84e6 - 2.21e6i)T + (3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-8.10e6 - 4.68e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (4.37e6 + 7.58e6i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-8.83e6 + 1.52e7i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 4.61e7iT - 6.45e14T^{2} \)
73 \( 1 + (1.80e7 - 3.12e7i)T + (-4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (2.53e7 + 4.39e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 3.31e7iT - 2.25e15T^{2} \)
89 \( 1 + (-7.90e7 + 4.56e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 1.64e8T + 7.83e15T^{2} \)
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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

−11.71561734659715779768457937496, −10.78655833959855705994730060428, −9.563579946926046582577371947000, −8.219562030105707342951854459039, −7.05587617043413425659622899896, −5.90452473129202782191831385121, −4.50834446311923585201307526307, −3.33764134245108771821578475028, −2.31651775416827189976409794758, −0.29674885767281686900845364977, 1.08671261914684263938631659056, 3.26950007256591825191302384616, 4.19183167235536708568226739224, 5.20582021840815427152632340235, 6.83209081708672787799425821955, 7.66100853116839473861256247766, 8.648435217238652195461011647565, 10.13758264411427608447960640100, 11.35002090151902155660525369931, 12.28447238508815560205348000095

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