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Properties

Label 2-126-63.11-c2-0-13
Degree $2$
Conductor $126$
Sign $-0.778 + 0.627i$
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 + (2.09 − 2.14i)3-s − 2.00·4-s + (−2.07 − 1.19i)5-s + (−3.03 − 2.96i)6-s + (−5.45 − 4.39i)7-s + 2.82i·8-s + (−0.213 − 8.99i)9-s + (−1.69 + 2.93i)10-s + (5.05 − 2.91i)11-s + (−4.19 + 4.29i)12-s + (−2.87 − 4.97i)13-s + (−6.21 + 7.70i)14-s + (−6.91 + 1.94i)15-s + 4.00·16-s + (21.4 + 12.3i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.698 − 0.715i)3-s − 0.500·4-s + (−0.414 − 0.239i)5-s + (−0.505 − 0.494i)6-s + (−0.778 − 0.627i)7-s + 0.353i·8-s + (−0.0236 − 0.999i)9-s + (−0.169 + 0.293i)10-s + (0.459 − 0.265i)11-s + (−0.349 + 0.357i)12-s + (−0.220 − 0.382i)13-s + (−0.443 + 0.550i)14-s + (−0.461 + 0.129i)15-s + 0.250·16-s + (1.26 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.778 + 0.627i$
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: \((2,\ 126,\ (\ :1),\ -0.778 + 0.627i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.456782 - 1.29502i\)
\(L(\frac12)\) \(\approx\) \(0.456782 - 1.29502i\)
\(L(2)\) 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 \( 1 + (-2.09 + 2.14i)T \)
7 \( 1 + (5.45 + 4.39i)T \)
good5 \( 1 + (2.07 + 1.19i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-5.05 + 2.91i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (2.87 + 4.97i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-21.4 - 12.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (1.06 + 1.85i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (0.123 + 0.0713i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-31.4 - 18.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 13.0T + 961T^{2} \)
37 \( 1 + (-16.0 - 27.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-27.3 + 15.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-40.1 + 69.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 61.1iT - 2.20e3T^{2} \)
53 \( 1 + (-22.5 - 13.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 22.7iT - 3.48e3T^{2} \)
61 \( 1 + 102.T + 3.72e3T^{2} \)
67 \( 1 - 70.2T + 4.48e3T^{2} \)
71 \( 1 + 100. iT - 5.04e3T^{2} \)
73 \( 1 + (4.36 - 7.56i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 62.2T + 6.24e3T^{2} \)
83 \( 1 + (51.5 + 29.7i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (94.6 - 54.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (79.8 - 138. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.51051784930472996582087113764, −12.15408439095128228978517676128, −10.59637445818530019996654403835, −9.613657904339714084439370794837, −8.475574358374744405731735882870, −7.50357304121071539559740109116, −6.14050740013769757501666642607, −4.06842449527598297052288635028, −2.98028604161267932885935846201, −0.957626283430187839694880644633, 2.97217801781594038620184860480, 4.28310437221268044800126286043, 5.71423611368662653510109583467, 7.14052145722108690474738990586, 8.202713733582254160657112503956, 9.385709459381481052677605897364, 9.922909261025160647496900518644, 11.52465587492414670527152811547, 12.65655822138100833438181701502, 13.91178234581498633381248968830

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