Properties

Label 2-126-7.3-c2-0-5
Degree $2$
Conductor $126$
Sign $-0.124 + 0.992i$
Analytic cond. $3.43325$
Root an. cond. $1.85290$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (1.24 − 0.717i)5-s + (1.74 − 6.77i)7-s + 2.82·8-s + (−1.75 − 1.01i)10-s + (3 − 5.19i)11-s − 21.3i·13-s + (−9.53 + 2.65i)14-s + (−2.00 − 3.46i)16-s + (7.75 + 4.47i)17-s + (−6.25 + 3.61i)19-s + 2.86i·20-s − 8.48·22-s + (−18.7 − 32.4i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.248 − 0.143i)5-s + (0.248 − 0.968i)7-s + 0.353·8-s + (−0.175 − 0.101i)10-s + (0.272 − 0.472i)11-s − 1.64i·13-s + (−0.681 + 0.189i)14-s + (−0.125 − 0.216i)16-s + (0.456 + 0.263i)17-s + (−0.329 + 0.190i)19-s + 0.143i·20-s − 0.385·22-s + (−0.814 − 1.41i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.124 + 0.992i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.124 + 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.765880 - 0.868003i\)
\(L(\frac12)\) \(\approx\) \(0.765880 - 0.868003i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 + (-1.74 + 6.77i)T \)
good5 \( 1 + (-1.24 + 0.717i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 21.3iT - 169T^{2} \)
17 \( 1 + (-7.75 - 4.47i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (6.25 - 3.61i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (18.7 + 32.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 33.9T + 841T^{2} \)
31 \( 1 + (-38.2 - 22.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.9 - 24.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 54.8iT - 1.68e3T^{2} \)
43 \( 1 + 1.48T + 1.84e3T^{2} \)
47 \( 1 + (-37.2 + 21.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (42.7 - 74.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-35.6 - 20.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.02 - 0.594i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.19 - 3.80i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 137.T + 5.04e3T^{2} \)
73 \( 1 + (-68.3 - 39.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (49.1 + 85.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 110. iT - 6.88e3T^{2} \)
89 \( 1 + (-18 + 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 10.9iT - 9.40e3T^{2} \)
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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

−12.82088244745319271077422135958, −11.78515110010294943280058679711, −10.46504043201182898801704681080, −10.16163796830884799266488554508, −8.539120188488608359804971261453, −7.76652422910699565943890142564, −6.16979816722841391061974085761, −4.55473651830101178572905839220, −3.08558533999374937300566535101, −0.984959416037264205730751098369, 2.06485288354938203844627194635, 4.38339910994110619223548300583, 5.79354444252777014471318938547, 6.79895035347936507592712923639, 8.097725127011049536803350414833, 9.198451460664972552486179676912, 9.928284491532883918820655639352, 11.49318976028026035918275408073, 12.21201397090609650641586979467, 13.79084171437968950307357398403

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