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Properties

Label 2-126-21.5-c1-0-0
Degree $2$
Conductor $126$
Sign $0.170 - 0.985i$
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 + (0.499 + 0.866i)4-s + (−2.09 + 3.62i)5-s + (−1.62 + 2.09i)7-s − 0.999i·8-s + (3.62 − 2.09i)10-s + (2.59 − 1.5i)11-s + 2.44i·13-s + (2.44 − 0.999i)14-s + (−0.5 + 0.866i)16-s + (0.507 + 0.878i)17-s + (−0.878 − 0.507i)19-s − 4.18·20-s − 3·22-s + (3.67 + 2.12i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.935 + 1.61i)5-s + (−0.612 + 0.790i)7-s − 0.353i·8-s + (1.14 − 0.661i)10-s + (0.783 − 0.452i)11-s + 0.679i·13-s + (0.654 − 0.267i)14-s + (−0.125 + 0.216i)16-s + (0.123 + 0.213i)17-s + (−0.201 − 0.116i)19-s − 0.935·20-s − 0.639·22-s + (0.766 + 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.170 - 0.985i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.170 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.475037 + 0.400088i\)
\(L(\frac12)\) \(\approx\) \(0.475037 + 0.400088i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (1.62 - 2.09i)T \)
good5 \( 1 + (2.09 - 3.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (-0.507 - 0.878i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.878 + 0.507i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.67 - 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.24iT - 29T^{2} \)
31 \( 1 + (-4.86 + 2.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.12 - 7.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.02T + 41T^{2} \)
43 \( 1 - 8.24T + 43T^{2} \)
47 \( 1 + (-0.507 + 0.878i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.07 - 0.621i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.76 - 9.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.12 - 2.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (-7.24 + 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.62 + 9.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.76iT - 97T^{2} \)
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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

−13.64022511100145126611757183351, −12.06721215045523466476678244052, −11.54670795550264709017258463135, −10.60728074113595806000538581018, −9.487021932915016855688267885036, −8.354647441621372978164072137576, −7.06681360570675071049674477177, −6.29688901508306406026988404175, −3.81142280534298255579098230675, −2.71963940280577965660453369776, 0.843184552907920420895455215135, 3.90737220084581918297396814081, 5.13840176340316216646983663888, 6.82347421749402705296418077361, 7.892068094955942799160412171864, 8.844460289182336456550097905087, 9.738111871103633978235974085116, 11.02875148301815762240006847626, 12.29651282198332153368508310886, 12.88376748228078509522518479993

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