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Properties

Label 2-126-63.16-c1-0-3
Degree $2$
Conductor $126$
Sign $0.176 - 0.984i$
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 + (0.796 + 1.53i)3-s + (−0.499 + 0.866i)4-s + 0.593·5-s + (−0.933 + 1.45i)6-s + (−0.0665 − 2.64i)7-s − 0.999·8-s + (−1.73 + 2.45i)9-s + (0.296 + 0.514i)10-s − 0.593·11-s + (−1.73 − 0.0789i)12-s + (−1.25 − 2.17i)13-s + (2.25 − 1.38i)14-s + (0.472 + 0.912i)15-s + (−0.5 − 0.866i)16-s + (1.46 + 2.52i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.460 + 0.887i)3-s + (−0.249 + 0.433i)4-s + 0.265·5-s + (−0.381 + 0.595i)6-s + (−0.0251 − 0.999i)7-s − 0.353·8-s + (−0.576 + 0.816i)9-s + (0.0938 + 0.162i)10-s − 0.178·11-s + (−0.499 − 0.0227i)12-s + (−0.348 − 0.603i)13-s + (0.603 − 0.368i)14-s + (0.122 + 0.235i)15-s + (−0.125 − 0.216i)16-s + (0.354 + 0.613i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.176 - 0.984i$
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: \((2,\ 126,\ (\ :1/2),\ 0.176 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06788 + 0.893678i\)
\(L(\frac12)\) \(\approx\) \(1.06788 + 0.893678i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.796 - 1.53i)T \)
7 \( 1 + (0.0665 + 2.64i)T \)
good5 \( 1 - 0.593T + 5T^{2} \)
11 \( 1 + 0.593T + 11T^{2} \)
13 \( 1 + (1.25 + 2.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.46 - 2.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.46T + 23T^{2} \)
29 \( 1 + (3.09 - 5.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.93 + 6.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.136 + 0.236i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.58 - 9.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.02 - 6.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.32 - 7.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.32 - 5.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.956 + 1.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (-3.95 - 6.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.62 - 8.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.85 + 6.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.21 - 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.86 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.63713897843092201878711345580, −13.11158060368776761513033123698, −11.41517244267444357624401895875, −10.32403771845386449736369598667, −9.450048195455622388443786700666, −8.170188508610662726987562730511, −7.17999232977680790648100722051, −5.56917180578508875854682927084, −4.45042070536359115715029558497, −3.14774410492995108083117912936, 1.91688645512099114379754115180, 3.22487293797384578041901895404, 5.24329848713899403295736960405, 6.40286948112792857949669210977, 7.82771127140044283043599271430, 9.037728217554841080456898795123, 9.895402757522100687027298169120, 11.60281267440460984855789408851, 12.09571545627447697428914422642, 13.09713122644268033309021157737

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