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Properties

Label 2-126-7.3-c8-0-15
Degree $2$
Conductor $126$
Sign $0.598 - 0.801i$
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  + (5.65 + 9.79i)2-s + (−63.9 + 110. i)4-s + (−664. + 383. i)5-s + (1.66e3 − 1.72e3i)7-s − 1.44e3·8-s + (−7.51e3 − 4.33e3i)10-s + (−2.52e3 + 4.37e3i)11-s − 2.82e4i·13-s + (2.63e4 + 6.56e3i)14-s + (−8.19e3 − 1.41e4i)16-s + (1.35e4 + 7.80e3i)17-s + (1.07e5 − 6.21e4i)19-s − 9.81e4i·20-s − 5.71e4·22-s + (−8.97e4 − 1.55e5i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−1.06 + 0.613i)5-s + (0.694 − 0.719i)7-s − 0.353·8-s + (−0.751 − 0.433i)10-s + (−0.172 + 0.298i)11-s − 0.990i·13-s + (0.686 + 0.170i)14-s + (−0.125 − 0.216i)16-s + (0.161 + 0.0934i)17-s + (0.825 − 0.476i)19-s − 0.613i·20-s − 0.243·22-s + (−0.320 − 0.555i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.598 - 0.801i$
Analytic conductor: \(51.3297\)
Root analytic conductor: \(7.16447\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :4),\ 0.598 - 0.801i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.994845463\)
\(L(\frac12)\) \(\approx\) \(1.994845463\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.65 - 9.79i)T \)
3 \( 1 \)
7 \( 1 + (-1.66e3 + 1.72e3i)T \)
good5 \( 1 + (664. - 383. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (2.52e3 - 4.37e3i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 + 2.82e4iT - 8.15e8T^{2} \)
17 \( 1 + (-1.35e4 - 7.80e3i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (-1.07e5 + 6.21e4i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (8.97e4 + 1.55e5i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + 7.11e4T + 5.00e11T^{2} \)
31 \( 1 + (-8.45e5 - 4.88e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-7.04e5 - 1.21e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 - 3.98e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.21e6T + 1.16e13T^{2} \)
47 \( 1 + (6.69e4 - 3.86e4i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-3.09e6 + 5.36e6i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (-1.27e7 - 7.37e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (2.14e6 - 1.23e6i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (1.79e6 - 3.10e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 - 3.51e7T + 6.45e14T^{2} \)
73 \( 1 + (-3.33e5 - 1.92e5i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (-1.68e7 - 2.91e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 7.92e7iT - 2.25e15T^{2} \)
89 \( 1 + (-4.51e7 + 2.60e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 1.44e8iT - 7.83e15T^{2} \)
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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

−11.93535406738998070114971536541, −11.07263670203140273460709686072, −10.01079416339235507178092758538, −8.244773607612474021837278883003, −7.66702006502726114195931050308, −6.72557005221555201529976846990, −5.16063555477267154636099553422, −4.08923322958005059066318934394, −2.94440317935982436109891208059, −0.75934069155233006886964396686, 0.75271132941041507690734203573, 2.11751661602913133287897639452, 3.68405736817615425968009541417, 4.68475485037008671499692956705, 5.78548241598423866889222469609, 7.56645450491362029373061056076, 8.544119789585592108190839341341, 9.523837206577462377484773664299, 11.01700028711978895115605403755, 11.86146585058047799442265191227

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