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Properties

Label 2-126-21.17-c1-0-3
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: $\chi_{126} (17, \cdot )$
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

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

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