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Properties

Label 2-126-21.11-c8-0-19
Degree $2$
Conductor $126$
Sign $-0.685 - 0.728i$
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 + (171. − 98.8i)5-s + (−1.59e3 − 1.79e3i)7-s − 1.44e3i·8-s + (1.11e3 − 1.93e3i)10-s + (1.66e3 + 962. i)11-s − 3.19e4·13-s + (−2.57e4 − 8.55e3i)14-s + (−8.19e3 − 1.41e4i)16-s + (2.75e4 + 1.59e4i)17-s + (−734. − 1.27e3i)19-s − 2.53e4i·20-s + 2.17e4·22-s + (−1.18e5 + 6.85e4i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.274 − 0.158i)5-s + (−0.664 − 0.747i)7-s − 0.353i·8-s + (0.111 − 0.193i)10-s + (0.113 + 0.0657i)11-s − 1.11·13-s + (−0.671 − 0.222i)14-s + (−0.125 − 0.216i)16-s + (0.330 + 0.190i)17-s + (−0.00563 − 0.00976i)19-s − 0.158i·20-s + 0.0929·22-s + (−0.424 + 0.244i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.728i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.685 - 0.728i$
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: \((2,\ 126,\ (\ :4),\ -0.685 - 0.728i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.1605435752\)
\(L(\frac12)\) \(\approx\) \(0.1605435752\)
\(L(5)\) 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 \( 1 + (1.59e3 + 1.79e3i)T \)
good5 \( 1 + (-171. + 98.8i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-1.66e3 - 962. i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 3.19e4T + 8.15e8T^{2} \)
17 \( 1 + (-2.75e4 - 1.59e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (734. + 1.27e3i)T + (-8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.18e5 - 6.85e4i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 1.50e5iT - 5.00e11T^{2} \)
31 \( 1 + (-1.20e5 + 2.08e5i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (8.99e5 + 1.55e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 - 2.54e6iT - 7.98e12T^{2} \)
43 \( 1 + 3.20e5T + 1.16e13T^{2} \)
47 \( 1 + (5.31e6 - 3.06e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (7.42e6 + 4.28e6i)T + (3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (6.20e6 + 3.58e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-2.97e6 - 5.14e6i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (1.20e7 - 2.08e7i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 2.30e7iT - 6.45e14T^{2} \)
73 \( 1 + (1.63e7 - 2.83e7i)T + (-4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-3.31e7 - 5.74e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + 4.15e7iT - 2.25e15T^{2} \)
89 \( 1 + (4.09e7 - 2.36e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 - 1.26e7T + 7.83e15T^{2} \)
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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

−11.25487254765967018036484571442, −10.09170960325188787073094071051, −9.457778220865771506797492578995, −7.71418170684255911089874333596, −6.64215697037408769999709225206, −5.41812598641491679762367116383, −4.19080805920249715957480059841, −3.00580912554622632551922291298, −1.54737532704464671960713488734, −0.03103863505992433748823086077, 2.15652315577356299794176271782, 3.24537054151087117084400532453, 4.76813119915088210434228836672, 5.88927902758976228201192092919, 6.82030369415705063922541537745, 8.079141987435418114527405417879, 9.359793799076197586099568541267, 10.30979321160636932210290026405, 11.86641045803076321528258031970, 12.39440951687114711960966751347

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