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Properties

Label 2-126-21.20-c9-0-21
Degree $2$
Conductor $126$
Sign $-0.409 + 0.912i$
Analytic cond. $64.8945$
Root an. cond. $8.05571$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16i·2-s − 256·4-s + 2.58e3·5-s + (5.46e3 − 3.23e3i)7-s + 4.09e3i·8-s − 4.14e4i·10-s − 1.08e4i·11-s − 8.70e4i·13-s + (−5.16e4 − 8.75e4i)14-s + 6.55e4·16-s − 3.15e5·17-s − 2.54e5i·19-s − 6.62e5·20-s − 1.73e5·22-s − 2.21e6i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.85·5-s + (0.861 − 0.508i)7-s + 0.353i·8-s − 1.30i·10-s − 0.223i·11-s − 0.844i·13-s + (−0.359 − 0.608i)14-s + 0.250·16-s − 0.916·17-s − 0.448i·19-s − 0.925·20-s − 0.158·22-s − 1.64i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.409 + 0.912i$
Analytic conductor: \(64.8945\)
Root analytic conductor: \(8.05571\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :9/2),\ -0.409 + 0.912i)\)

Particular Values

\(L(5)\) \(\approx\) \(3.099468454\)
\(L(\frac12)\) \(\approx\) \(3.099468454\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 16iT \)
3 \( 1 \)
7 \( 1 + (-5.46e3 + 3.23e3i)T \)
good5 \( 1 - 2.58e3T + 1.95e6T^{2} \)
11 \( 1 + 1.08e4iT - 2.35e9T^{2} \)
13 \( 1 + 8.70e4iT - 1.06e10T^{2} \)
17 \( 1 + 3.15e5T + 1.18e11T^{2} \)
19 \( 1 + 2.54e5iT - 3.22e11T^{2} \)
23 \( 1 + 2.21e6iT - 1.80e12T^{2} \)
29 \( 1 - 6.16e6iT - 1.45e13T^{2} \)
31 \( 1 - 1.08e5iT - 2.64e13T^{2} \)
37 \( 1 + 6.02e6T + 1.29e14T^{2} \)
41 \( 1 - 2.25e7T + 3.27e14T^{2} \)
43 \( 1 - 3.01e6T + 5.02e14T^{2} \)
47 \( 1 - 2.96e7T + 1.11e15T^{2} \)
53 \( 1 + 6.66e7iT - 3.29e15T^{2} \)
59 \( 1 + 1.22e8T + 8.66e15T^{2} \)
61 \( 1 + 4.44e7iT - 1.16e16T^{2} \)
67 \( 1 + 9.86e7T + 2.72e16T^{2} \)
71 \( 1 - 1.25e7iT - 4.58e16T^{2} \)
73 \( 1 - 2.26e8iT - 5.88e16T^{2} \)
79 \( 1 + 3.89e8T + 1.19e17T^{2} \)
83 \( 1 - 7.14e8T + 1.86e17T^{2} \)
89 \( 1 - 6.11e8T + 3.50e17T^{2} \)
97 \( 1 - 5.32e8iT - 7.60e17T^{2} \)
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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

−10.77580427451179831566251269473, −10.61024682441408608643231195758, −9.325864105479797985061862926360, −8.463408772361086990393190600546, −6.79583544843602048282752533077, −5.54177678162257947961382230413, −4.60401488551105639039021254670, −2.79465095369226397413417346718, −1.83025906263101897154371694868, −0.75249072176332067080666906543, 1.45603098944665637699044102547, 2.32546731196866729285112954029, 4.46180966346132848897193704097, 5.58701130336499669514329939083, 6.25983111700348398280017107947, 7.58558045346682014714679394044, 9.013141463732171749366602188051, 9.494687096438685751657877739135, 10.74564699220345917653012537429, 12.03946363804702337592365971454

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