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Properties

Label 2-126-7.5-c8-0-9
Degree $2$
Conductor $126$
Sign $0.935 - 0.353i$
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 + (565. + 326. i)5-s + (2.12e3 − 1.12e3i)7-s − 1.44e3·8-s + (6.40e3 − 3.69e3i)10-s + (1.38e4 + 2.40e4i)11-s + 6.76e3i·13-s + (965. − 2.71e4i)14-s + (−8.19e3 + 1.41e4i)16-s + (−8.45e4 + 4.88e4i)17-s + (6.77e4 + 3.90e4i)19-s − 8.36e4i·20-s + 3.14e5·22-s + (−4.68e4 + 8.11e4i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.905 + 0.522i)5-s + (0.883 − 0.468i)7-s − 0.353·8-s + (0.640 − 0.369i)10-s + (0.948 + 1.64i)11-s + 0.236i·13-s + (0.0251 − 0.706i)14-s + (−0.125 + 0.216i)16-s + (−1.01 + 0.584i)17-s + (0.519 + 0.299i)19-s − 0.522i·20-s + 1.34·22-s + (−0.167 + 0.289i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.935 - 0.353i$
Analytic conductor: \(51.3297\)
Root analytic conductor: \(7.16447\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :4),\ 0.935 - 0.353i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.147454659\)
\(L(\frac12)\) \(\approx\) \(3.147454659\)
\(L(5)\) 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 \( 1 + (-2.12e3 + 1.12e3i)T \)
good5 \( 1 + (-565. - 326. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-1.38e4 - 2.40e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 6.76e3iT - 8.15e8T^{2} \)
17 \( 1 + (8.45e4 - 4.88e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-6.77e4 - 3.90e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (4.68e4 - 8.11e4i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 3.05e5T + 5.00e11T^{2} \)
31 \( 1 + (1.10e6 - 6.38e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (1.39e6 - 2.41e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 2.12e6iT - 7.98e12T^{2} \)
43 \( 1 - 5.96e6T + 1.16e13T^{2} \)
47 \( 1 + (-3.70e6 - 2.14e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-1.11e6 - 1.92e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-1.57e7 + 9.08e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (4.27e6 + 2.47e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (1.58e7 + 2.75e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 2.29e7T + 6.45e14T^{2} \)
73 \( 1 + (1.88e6 - 1.08e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (7.82e6 - 1.35e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 1.88e7iT - 2.25e15T^{2} \)
89 \( 1 + (-6.59e7 - 3.80e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
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

−11.84039516486829766699455091128, −10.83681586110599067635299737195, −9.970948983616065405013353924023, −9.058504434267650593164338951237, −7.38756536338405470971874021618, −6.33644969110718626297702215613, −4.91595650389698624034669240813, −3.90875431613423694722433364670, −2.12307600183870381457130379487, −1.48637932454997210846190154230, 0.74214843615962198751785799684, 2.23955283782419288235699946628, 3.93023312634407704544438487613, 5.37636707802036269596137335899, 5.91014504696638488096474567064, 7.35704127506075494952243363602, 8.850606468585121216522724712795, 9.066007298860121831063780721178, 10.92907731584179950713925593398, 11.78468976804499041761042303218

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