Properties

Label 2-126-63.16-c1-0-0
Degree $2$
Conductor $126$
Sign $-0.778 - 0.627i$
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 + (−1.09 + 1.34i)3-s + (−0.499 + 0.866i)4-s − 3.18·5-s + (−1.71 − 0.272i)6-s + (0.710 + 2.54i)7-s − 0.999·8-s + (−0.619 − 2.93i)9-s + (−1.59 − 2.75i)10-s + 3.18·11-s + (−0.619 − 1.61i)12-s + (2.85 + 4.93i)13-s + (−1.85 + 1.88i)14-s + (3.47 − 4.28i)15-s + (−0.5 − 0.866i)16-s + (−0.760 − 1.31i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.629 + 0.776i)3-s + (−0.249 + 0.433i)4-s − 1.42·5-s + (−0.698 − 0.111i)6-s + (0.268 + 0.963i)7-s − 0.353·8-s + (−0.206 − 0.978i)9-s + (−0.503 − 0.871i)10-s + 0.959·11-s + (−0.178 − 0.466i)12-s + (0.790 + 1.36i)13-s + (−0.494 + 0.505i)14-s + (0.896 − 1.10i)15-s + (−0.125 − 0.216i)16-s + (−0.184 − 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.778 - 0.627i$
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: \((2,\ 126,\ (\ :1/2),\ -0.778 - 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273215 + 0.774285i\)
\(L(\frac12)\) \(\approx\) \(0.273215 + 0.774285i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 - 1.34i)T \)
7 \( 1 + (-0.710 - 2.54i)T \)
good5 \( 1 + 3.18T + 5T^{2} \)
11 \( 1 - 3.18T + 11T^{2} \)
13 \( 1 + (-2.85 - 4.93i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.760 + 1.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.641 - 1.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.23T + 23T^{2} \)
29 \( 1 + (3.54 - 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.71 + 8.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.80 + 4.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.41 + 5.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.02 - 1.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.562 + 0.974i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.56 + 2.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.48 - 9.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 + (2.48 + 4.30i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.03 - 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.112 + 0.195i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.42 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.13634867592788218311233677310, −12.42750116881381981572973108615, −11.66859255649554420025152577541, −11.18097526044979165918806812686, −9.252161766511034343310883989937, −8.598386425988124166028155274702, −7.07452238466565773544418706421, −5.95721394063676308002915931262, −4.54029461108714216190735535763, −3.72482347751956533205239016958, 0.933624983337636158523858647482, 3.51011320438515015745342506711, 4.69393714370206280513870960388, 6.33901154360039921833016422647, 7.53950574370617436301038422251, 8.441813217823046133040286222695, 10.41536900339418581714976708502, 11.19259231898487922115964248667, 11.84862307226971439691431353380, 12.84749584096421214894120455404

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