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Properties

Label 2-126-63.11-c2-0-6
Degree $2$
Conductor $126$
Sign $-0.103 - 0.994i$
Analytic cond. $3.43325$
Root an. cond. $1.85290$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (2.25 + 1.98i)3-s − 2.00·4-s + (2.56 + 1.48i)5-s + (−2.80 + 3.18i)6-s + (6.84 − 1.47i)7-s − 2.82i·8-s + (1.15 + 8.92i)9-s + (−2.09 + 3.62i)10-s + (−9.06 + 5.23i)11-s + (−4.50 − 3.96i)12-s + (−1.61 − 2.79i)13-s + (2.09 + 9.67i)14-s + (2.84 + 8.41i)15-s + 4.00·16-s + (−4.26 − 2.46i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.751 + 0.660i)3-s − 0.500·4-s + (0.512 + 0.296i)5-s + (−0.466 + 0.531i)6-s + (0.977 − 0.211i)7-s − 0.353i·8-s + (0.128 + 0.991i)9-s + (−0.209 + 0.362i)10-s + (−0.824 + 0.475i)11-s + (−0.375 − 0.330i)12-s + (−0.124 − 0.214i)13-s + (0.149 + 0.691i)14-s + (0.189 + 0.561i)15-s + 0.250·16-s + (−0.251 − 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.103 - 0.994i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.103 - 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.22960 + 1.36372i\)
\(L(\frac12)\) \(\approx\) \(1.22960 + 1.36372i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (-2.25 - 1.98i)T \)
7 \( 1 + (-6.84 + 1.47i)T \)
good5 \( 1 + (-2.56 - 1.48i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (9.06 - 5.23i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (1.61 + 2.79i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (4.26 + 2.46i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (10.3 + 17.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-12.7 - 7.38i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-26.9 - 15.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 26.4T + 961T^{2} \)
37 \( 1 + (25.0 + 43.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-42.6 + 24.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-26.9 + 46.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 19.4iT - 2.20e3T^{2} \)
53 \( 1 + (86.9 + 50.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 27.8iT - 3.48e3T^{2} \)
61 \( 1 + 40.6T + 3.72e3T^{2} \)
67 \( 1 - 76.1T + 4.48e3T^{2} \)
71 \( 1 + 2.51iT - 5.04e3T^{2} \)
73 \( 1 + (69.1 - 119. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 32.6T + 6.24e3T^{2} \)
83 \( 1 + (-7.18 - 4.14i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (107. - 61.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-44.1 + 76.4i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.84960935781301621467113696136, −12.74874763097081180906228206447, −10.97136467216752193629585667733, −10.19778225435137571903470381819, −9.038384418221123495594821751836, −8.078610259051600255602907464323, −7.06962297079725309204054792774, −5.33621366641427666236059637813, −4.39885907979023841072225434454, −2.47786353182935790252666397280, 1.49582443396562702126147886024, 2.79625494551732291526527245330, 4.60552096637533123591311005803, 6.09859744944396689977534408357, 7.86831948110257127020827558720, 8.538390117884773339439187066938, 9.666247378180425060666591991307, 10.86092481709750262662538795530, 11.99021221155204069706798602555, 12.90247971785288523070148011036

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