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Properties

Label 2-126-63.20-c1-0-0
Degree $2$
Conductor $126$
Sign $0.999 - 0.0248i$
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  + (−0.866 − 0.5i)2-s + (−1.62 − 0.608i)3-s + (0.499 + 0.866i)4-s + (1.94 + 3.36i)5-s + (1.10 + 1.33i)6-s + (0.343 − 2.62i)7-s − 0.999i·8-s + (2.26 + 1.97i)9-s − 3.89i·10-s + (3.41 + 1.97i)11-s + (−0.284 − 1.70i)12-s + (2.46 − 1.42i)13-s + (−1.60 + 2.09i)14-s + (−1.10 − 6.64i)15-s + (−0.5 + 0.866i)16-s + 0.742·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.936 − 0.351i)3-s + (0.249 + 0.433i)4-s + (0.870 + 1.50i)5-s + (0.449 + 0.546i)6-s + (0.130 − 0.991i)7-s − 0.353i·8-s + (0.753 + 0.657i)9-s − 1.23i·10-s + (1.03 + 0.594i)11-s + (−0.0820 − 0.493i)12-s + (0.684 − 0.395i)13-s + (−0.430 + 0.561i)14-s + (−0.285 − 1.71i)15-s + (−0.125 + 0.216i)16-s + 0.179·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.999 - 0.0248i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.999 - 0.0248i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.750199 + 0.00932956i\)
\(L(\frac12)\) \(\approx\) \(0.750199 + 0.00932956i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (1.62 + 0.608i)T \)
7 \( 1 + (-0.343 + 2.62i)T \)
good5 \( 1 + (-1.94 - 3.36i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.41 - 1.97i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.46 + 1.42i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.742T + 17T^{2} \)
19 \( 1 - 1.78iT - 19T^{2} \)
23 \( 1 + (5.41 - 3.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.50 + 1.44i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.04 + 1.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + (5.24 + 9.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.471 + 0.816i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.09 + 1.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-0.0105 - 0.0183i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.13 + 1.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.72 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + 4.85iT - 73T^{2} \)
79 \( 1 + (1.81 - 3.14i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.02 - 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.26T + 89T^{2} \)
97 \( 1 + (-16.2 - 9.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.42761497485424215861472971553, −12.03148180797490317427074474290, −11.10332375086108912893841769882, −10.35950464890625385038512553654, −9.719839460439537108343983625116, −7.69589373783946846158882301430, −6.82502976983418647236598135450, −5.97152323443471734815770992978, −3.80657896293515489137829388096, −1.75286504852850702134312639533, 1.38163171314482238801025731453, 4.55821505934178441290975022628, 5.74561189886145874605862638075, 6.33810667302022386337142370677, 8.474873766952050728591281778610, 9.117288328719573924689136939063, 9.960242186015325856153193377385, 11.42386685551071987505501456309, 12.14071709156131727455148873117, 13.22671002948764987163603830750

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