Properties

Label 2-126-7.2-c7-0-18
Degree $2$
Conductor $126$
Sign $0.581 + 0.813i$
Analytic cond. $39.3605$
Root an. cond. $6.27379$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 + 6.92i)2-s + (−31.9 − 55.4i)4-s + (212. − 367. i)5-s + (−30.7 − 906. i)7-s + 511.·8-s + (1.69e3 + 2.93e3i)10-s + (3.94e3 + 6.83e3i)11-s + 6.71e3·13-s + (6.40e3 + 3.41e3i)14-s + (−2.04e3 + 3.54e3i)16-s + (−3.53e3 − 6.12e3i)17-s + (1.31e4 − 2.27e4i)19-s − 2.71e4·20-s − 6.31e4·22-s + (6.03e3 − 1.04e4i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.758 − 1.31i)5-s + (−0.0338 − 0.999i)7-s + 0.353·8-s + (0.536 + 0.929i)10-s + (0.893 + 1.54i)11-s + 0.848·13-s + (0.623 + 0.332i)14-s + (−0.125 + 0.216i)16-s + (−0.174 − 0.302i)17-s + (0.438 − 0.760i)19-s − 0.758·20-s − 1.26·22-s + (0.103 − 0.179i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(39.3605\)
Root analytic conductor: \(6.27379\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :7/2),\ 0.581 + 0.813i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.998463592\)
\(L(\frac12)\) \(\approx\) \(1.998463592\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4 - 6.92i)T \)
3 \( 1 \)
7 \( 1 + (30.7 + 906. i)T \)
good5 \( 1 + (-212. + 367. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
11 \( 1 + (-3.94e3 - 6.83e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 - 6.71e3T + 6.27e7T^{2} \)
17 \( 1 + (3.53e3 + 6.12e3i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-1.31e4 + 2.27e4i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (-6.03e3 + 1.04e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + 3.06e3T + 1.72e10T^{2} \)
31 \( 1 + (-5.92e4 - 1.02e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (-2.29e5 + 3.98e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + 3.16e5T + 1.94e11T^{2} \)
43 \( 1 + 3.16e4T + 2.71e11T^{2} \)
47 \( 1 + (-4.03e5 + 6.98e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (-2.39e5 - 4.15e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-3.37e5 - 5.84e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-2.83e5 + 4.90e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (5.92e5 + 1.02e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + 4.61e6T + 9.09e12T^{2} \)
73 \( 1 + (1.52e6 + 2.63e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-3.45e6 + 5.97e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 - 9.01e6T + 2.71e13T^{2} \)
89 \( 1 + (3.50e6 - 6.07e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 8.60e6T + 8.07e13T^{2} \)
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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

−12.00569363065519121788268126066, −10.50024097150007022609841495238, −9.473009801783980639769157710777, −8.874628981113833399414696961364, −7.46674334760732718818711471659, −6.48513799198602969893874127221, −5.07663771282332683277123990818, −4.19370296260807928982347510408, −1.67430820373114680327928779499, −0.72849297592262513352046855995, 1.32664873855585998892525541528, 2.69522305804815922169809783816, 3.58863433550612854807905531198, 5.81945056687442256532136741687, 6.45881752273243261451555126526, 8.192889625861135033333754989084, 9.143943304733209861727597397998, 10.17296866427299533858107377696, 11.19003269344061262998380898611, 11.76927906487597291460900377026

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