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Properties

Label 2-126-7.5-c8-0-19
Degree $2$
Conductor $126$
Sign $-0.471 + 0.881i$
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 + (758. + 437. i)5-s + (855. − 2.24e3i)7-s − 1.44e3·8-s + (8.57e3 − 4.95e3i)10-s + (−6.41e3 − 1.11e4i)11-s − 3.06e3i·13-s + (−1.71e4 − 2.10e4i)14-s + (−8.19e3 + 1.41e4i)16-s + (−6.24e4 + 3.60e4i)17-s + (4.90e4 + 2.83e4i)19-s − 1.12e5i·20-s − 1.45e5·22-s + (2.49e5 − 4.32e5i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (1.21 + 0.700i)5-s + (0.356 − 0.934i)7-s − 0.353·8-s + (0.857 − 0.495i)10-s + (−0.438 − 0.758i)11-s − 0.107i·13-s + (−0.446 − 0.548i)14-s + (−0.125 + 0.216i)16-s + (−0.747 + 0.431i)17-s + (0.376 + 0.217i)19-s − 0.700i·20-s − 0.619·22-s + (0.891 − 1.54i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.471 + 0.881i$
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: \((2,\ 126,\ (\ :4),\ -0.471 + 0.881i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.794595531\)
\(L(\frac12)\) \(\approx\) \(2.794595531\)
\(L(5)\) 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 \( 1 + (-855. + 2.24e3i)T \)
good5 \( 1 + (-758. - 437. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (6.41e3 + 1.11e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 3.06e3iT - 8.15e8T^{2} \)
17 \( 1 + (6.24e4 - 3.60e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-4.90e4 - 2.83e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-2.49e5 + 4.32e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 8.83e4T + 5.00e11T^{2} \)
31 \( 1 + (-5.01e5 + 2.89e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-4.65e5 + 8.05e5i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 1.42e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.95e6T + 1.16e13T^{2} \)
47 \( 1 + (6.92e6 + 3.99e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (1.04e6 + 1.80e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (7.60e6 - 4.38e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (1.25e7 + 7.27e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-7.05e6 - 1.22e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 3.34e7T + 6.45e14T^{2} \)
73 \( 1 + (-1.12e6 + 6.47e5i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-1.69e7 + 2.93e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 7.65e6iT - 2.25e15T^{2} \)
89 \( 1 + (8.82e7 + 5.09e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 2.91e7iT - 7.83e15T^{2} \)
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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

−11.10541961718543067761236627253, −10.63918318519600547096662711542, −9.755704169659099340504622110457, −8.413524130510190392263053671311, −6.85932902168297627328946626997, −5.87251974284011389298612683817, −4.57756502877845612922438216573, −3.12012469958349985900307086800, −2.01588949162979932006981565848, −0.63757313253507804174950402851, 1.48441523885102451863380165294, 2.71565617502287762812657906137, 4.78629538675776902206177803118, 5.36698665784573022612223678308, 6.49873368290430781085373322483, 7.85458462579748219845720247472, 9.114670513122410383525618229876, 9.623412568451209219644178009581, 11.29575247669031139153875867360, 12.46116632011946140821236487465

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