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Properties

Label 2-126-63.59-c1-0-4
Degree $2$
Conductor $126$
Sign $0.639 - 0.768i$
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 + (1.47 + 0.914i)3-s + (0.499 + 0.866i)4-s − 3.64·5-s + (0.816 + 1.52i)6-s + (2.62 − 0.310i)7-s + 0.999i·8-s + (1.32 + 2.69i)9-s + (−3.15 − 1.82i)10-s − 5.06i·11-s + (−0.0569 + 1.73i)12-s + (−2.94 − 1.69i)13-s + (2.43 + 1.04i)14-s + (−5.35 − 3.33i)15-s + (−0.5 + 0.866i)16-s + (0.774 − 1.34i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.849 + 0.528i)3-s + (0.249 + 0.433i)4-s − 1.62·5-s + (0.333 + 0.623i)6-s + (0.993 − 0.117i)7-s + 0.353i·8-s + (0.442 + 0.897i)9-s + (−0.997 − 0.576i)10-s − 1.52i·11-s + (−0.0164 + 0.499i)12-s + (−0.816 − 0.471i)13-s + (0.649 + 0.279i)14-s + (−1.38 − 0.860i)15-s + (−0.125 + 0.216i)16-s + (0.187 − 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.639 - 0.768i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.639 - 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41866 + 0.665010i\)
\(L(\frac12)\) \(\approx\) \(1.41866 + 0.665010i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.47 - 0.914i)T \)
7 \( 1 + (-2.62 + 0.310i)T \)
good5 \( 1 + 3.64T + 5T^{2} \)
11 \( 1 + 5.06iT - 11T^{2} \)
13 \( 1 + (2.94 + 1.69i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.774 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.707 - 0.408i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.70iT - 23T^{2} \)
29 \( 1 + (3.60 - 2.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.87 - 1.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.39 + 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.01 - 1.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.06 - 5.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.37 - 5.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.4 - 6.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.08 + 1.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.28 + 3.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.22 + 2.12i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.74iT - 71T^{2} \)
73 \( 1 + (-3.76 - 2.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.37 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.768 + 1.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.01 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.59 + 3.23i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.85689936966850627207672106272, −12.56025617502110053690928043530, −11.42965434934570852915578555395, −10.77554231922570205341143264284, −8.898319876376606163538584138700, −7.972650851285790837765732792683, −7.44684538814746305754720510279, −5.27806904062822855404803432283, −4.14189421612921891316483862704, −3.13491137532021259391240709897, 2.10701847212238952824913285405, 3.86648455720155175559569081046, 4.75505994491024994992547366070, 7.06478105072999636672117009336, 7.65782708149675466867591889358, 8.795373547611651159026115833279, 10.25276103534752830764308965719, 11.78590942108347222557425706977, 12.04280767300451552702434494162, 13.08454739359144454839476082888

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