Properties

Label 2-126-9.4-c1-0-4
Degree $2$
Conductor $126$
Sign $0.939 + 0.342i$
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 − 1.73i·3-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s + (1.49 − 0.866i)6-s + (−0.5 − 0.866i)7-s − 0.999·8-s − 2.99·9-s + 3·10-s + (3 + 5.19i)11-s + (1.49 + 0.866i)12-s + (−1 + 1.73i)13-s + (0.499 − 0.866i)14-s + (−4.5 − 2.59i)15-s + (−0.5 − 0.866i)16-s + 6·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 0.999i·3-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s + (0.612 − 0.353i)6-s + (−0.188 − 0.327i)7-s − 0.353·8-s − 0.999·9-s + 0.948·10-s + (0.904 + 1.56i)11-s + (0.433 + 0.249i)12-s + (−0.277 + 0.480i)13-s + (0.133 − 0.231i)14-s + (−1.16 − 0.670i)15-s + (−0.125 − 0.216i)16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28028 - 0.225747i\)
\(L(\frac12)\) \(\approx\) \(1.28028 - 0.225747i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.20986910823196139391488959493, −12.54574775076249188563250726737, −11.88455747256029342077452353981, −9.865036300491147798837126901256, −8.943665352280116993837185077328, −7.72947248319515276352626506245, −6.71548736137882161307211737010, −5.61153587545137261138159368982, −4.28245679157121827719439111253, −1.77013939322723170388853964880, 2.78406084968574424703432640976, 3.75638913511956709659578632254, 5.58530102299595835378930166780, 6.34169381378257625831735145687, 8.475795705608970126586706196731, 9.578776388011096610624079001792, 10.52514854254777462354520007841, 11.06723454414901334002775045985, 12.24928004782496096231071561576, 13.65357224935547719695164869503

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