Properties

Label 2-126-63.16-c1-0-1
Degree $2$
Conductor $126$
Sign $0.932 - 0.360i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.619 + 1.61i)3-s + (−0.499 + 0.866i)4-s + 1.76·5-s + (1.09 − 1.34i)6-s + (−1.85 + 1.88i)7-s + 0.999·8-s + (−2.23 + 2.00i)9-s + (−0.880 − 1.52i)10-s + 6.12·11-s + (−1.71 − 0.272i)12-s + (−0.380 − 0.658i)13-s + (2.56 + 0.658i)14-s + (1.09 + 2.84i)15-s + (−0.5 − 0.866i)16-s + (−3.42 − 5.92i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.357 + 0.933i)3-s + (−0.249 + 0.433i)4-s + 0.787·5-s + (0.445 − 0.549i)6-s + (−0.699 + 0.714i)7-s + 0.353·8-s + (−0.744 + 0.668i)9-s + (−0.278 − 0.482i)10-s + 1.84·11-s + (−0.493 − 0.0785i)12-s + (−0.105 − 0.182i)13-s + (0.684 + 0.176i)14-s + (0.281 + 0.735i)15-s + (−0.125 − 0.216i)16-s + (−0.829 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.932 - 0.360i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.932 - 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03286 + 0.192623i\)
\(L(\frac12)\) \(\approx\) \(1.03286 + 0.192623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.619 - 1.61i)T \)
7 \( 1 + (1.85 - 1.88i)T \)
good5 \( 1 - 1.76T + 5T^{2} \)
11 \( 1 - 6.12T + 11T^{2} \)
13 \( 1 + (0.380 + 0.658i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.42 + 5.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.971 + 1.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.421T + 23T^{2} \)
29 \( 1 + (-0.732 + 1.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.85 - 6.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.44 + 2.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.47 + 6.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 7.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.830 + 1.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.112 + 0.195i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.993 - 1.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.17 - 8.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.39 - 5.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + (-0.153 - 0.265i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.72 - 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.56 - 2.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.30 + 2.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.81 - 3.14i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.58241344239110627243981851791, −12.17876662681443053718114506039, −11.32250453452765251182154822687, −10.05829639017993483027583967290, −9.205997770174821808102695185454, −8.914528273159782645156597601528, −6.84586009112041544798600984082, −5.38156402585809942558652171557, −3.82602350489586755864695107132, −2.43264635136228650574994170717, 1.60239157221177772127524824236, 3.88873377960504604175660913491, 6.24704939071819496207087011450, 6.49010363556043403578133744722, 7.84300079451332610282720966480, 9.094256753027366420301790684643, 9.772892219903700947753601135788, 11.26849199028467323822118719503, 12.62213775704383326477558567447, 13.47617590443718861434047158303

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