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Properties

Label 2-126-63.41-c1-0-0
Degree $2$
Conductor $126$
Sign $-0.905 - 0.424i$
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 + (−1.40 − 1.00i)3-s + (0.499 − 0.866i)4-s + (−1.17 + 2.03i)5-s + (1.72 + 0.167i)6-s + (−2.63 + 0.274i)7-s + 0.999i·8-s + (0.971 + 2.83i)9-s − 2.34i·10-s + (−4.91 + 2.83i)11-s + (−1.57 + 0.716i)12-s + (−1.48 − 0.859i)13-s + (2.14 − 1.55i)14-s + (3.70 − 1.68i)15-s + (−0.5 − 0.866i)16-s + 1.76·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.813 − 0.581i)3-s + (0.249 − 0.433i)4-s + (−0.525 + 0.909i)5-s + (0.703 + 0.0684i)6-s + (−0.994 + 0.103i)7-s + 0.353i·8-s + (0.323 + 0.946i)9-s − 0.742i·10-s + (−1.48 + 0.855i)11-s + (−0.455 + 0.206i)12-s + (−0.413 − 0.238i)13-s + (0.572 − 0.415i)14-s + (0.956 − 0.434i)15-s + (−0.125 − 0.216i)16-s + 0.429·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.905 - 0.424i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ -0.905 - 0.424i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0434084 + 0.194640i\)
\(L(\frac12)\) \(\approx\) \(0.0434084 + 0.194640i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 + 1.00i)T \)
7 \( 1 + (2.63 - 0.274i)T \)
good5 \( 1 + (1.17 - 2.03i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.91 - 2.83i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.48 + 0.859i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.76T + 17T^{2} \)
19 \( 1 + 1.13iT - 19T^{2} \)
23 \( 1 + (3.18 + 1.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.59 + 2.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.24 - 4.18i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.19T + 37T^{2} \)
41 \( 1 + (3.99 - 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.76 - 3.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.90 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (1.11 - 1.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.79 - 4.49i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.43 - 9.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.52iT - 71T^{2} \)
73 \( 1 - 5.34iT - 73T^{2} \)
79 \( 1 + (-6.51 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.27 - 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.16T + 89T^{2} \)
97 \( 1 + (-3.97 + 2.29i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.71555750683945101697581566718, −12.62194268436439589342575975414, −11.77594467090928956514440129095, −10.37414335886544595867633600007, −10.13104919330099842811274663363, −8.138168719301450788555931137647, −7.21690335231912713807459261730, −6.46893420539124022764167412304, −5.09031975732348170934515636376, −2.71446250509561120252557239131, 0.26560947720268072084386099054, 3.34815640822186559098790816227, 4.86047369304104424223478689283, 6.16006089334416266548260275174, 7.73216671853846231045924816450, 8.867604966544142511019852310193, 9.990554347935237917471840350725, 10.66307462170377909416233187961, 11.98605370695281803013882819842, 12.49140712361171081515937980335

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