Properties

Degree $2$
Conductor $126$
Sign $-0.00922 + 0.999i$
Motivic weight $1$
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 − 1.41i)3-s + (−0.499 + 0.866i)4-s + (0.724 − 1.25i)5-s + (−1.72 − 0.158i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−1.00 − 2.82i)9-s − 1.44·10-s + (−1 − 1.73i)11-s + (0.724 + 1.57i)12-s + (−2.44 + 4.24i)13-s + (0.499 − 0.866i)14-s + (−1.05 − 2.28i)15-s + (−0.5 − 0.866i)16-s + 2·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.577 − 0.816i)3-s + (−0.249 + 0.433i)4-s + (0.324 − 0.561i)5-s + (−0.704 − 0.0648i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.333 − 0.942i)9-s − 0.458·10-s + (−0.301 − 0.522i)11-s + (0.209 + 0.454i)12-s + (−0.679 + 1.17i)13-s + (0.133 − 0.231i)14-s + (−0.271 − 0.588i)15-s + (−0.125 − 0.216i)16-s + 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00922 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00922 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.00922 + 0.999i$
Motivic weight: \(1\)
Character: $\chi_{126} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ -0.00922 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755770 - 0.762777i\)
\(L(\frac12)\) \(\approx\) \(0.755770 - 0.762777i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.41i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.724 + 1.25i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.44 - 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.44 - 5.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 + (4.89 - 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.44 + 5.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.89 + 8.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.27 + 5.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.44 + 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.101T + 71T^{2} \)
73 \( 1 + 6.89T + 73T^{2} \)
79 \( 1 + (-0.949 - 1.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + (1.44 + 2.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.97280171581791195257764470298, −12.18434640722822679912361872418, −11.28805785214353737047531727260, −9.709702493442751698125376658079, −8.934326664491490091346343189615, −7.998598881879335966217129888521, −6.73378771462966882349478729921, −5.05631759602413752567637533827, −3.12010981632213371051047079961, −1.58132965029874222363763505206, 2.79404569706707804569198721999, 4.56265206948032817000657907568, 5.79538136237070114289267443347, 7.44229396240704930344692185736, 8.148890594196736782438973583706, 9.725615955852598893909644741957, 10.06587252097080663284932025638, 11.20557514430857367363017835591, 12.91010140351248478712902324930, 14.01333259215675992935018739403

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