Properties

Label 2-126-9.4-c1-0-3
Degree $2$
Conductor $126$
Sign $0.766 - 0.642i$
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 + (1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−1 + 1.73i)5-s + (1.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 0.999·8-s + (1.5 − 2.59i)9-s − 1.99·10-s + (−0.5 − 0.866i)11-s + 1.73i·12-s + (3 − 5.19i)13-s + (−0.499 + 0.866i)14-s + 3.46i·15-s + (−0.5 − 0.866i)16-s − 5·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.447 + 0.774i)5-s + (0.612 + 0.353i)6-s + (0.188 + 0.327i)7-s − 0.353·8-s + (0.5 − 0.866i)9-s − 0.632·10-s + (−0.150 − 0.261i)11-s + 0.499i·12-s + (0.832 − 1.44i)13-s + (−0.133 + 0.231i)14-s + 0.894i·15-s + (−0.125 − 0.216i)16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.766 - 0.642i$
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: \((2,\ 126,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37693 + 0.501164i\)
\(L(\frac12)\) \(\approx\) \(1.37693 + 0.501164i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8 + 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.39560702532095048814899539897, −12.97155915050280027369317266507, −11.56602850860440395960291097433, −10.42400442228366273439856855010, −8.756128605346644803185162649711, −8.112813899750194172499269198264, −7.00045178256891357249556825515, −5.95044321299502134099436925272, −4.01045342884639869263048573210, −2.73566829821513854369043767683, 2.12785420891147681655508486085, 4.14404042175073924164565894554, 4.52480945873364556384129355616, 6.64908300524279367363703606192, 8.441339527570770070614362827090, 8.876711856484510325326696871937, 10.24557188885592374799801775324, 11.15660330935040008044317610864, 12.34812941571619881299164716439, 13.34847133560803116986308430748

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