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Properties

Label 2-126-63.23-c2-0-0
Degree $2$
Conductor $126$
Sign $-0.844 - 0.535i$
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  − 1.41i·2-s + (−0.664 + 2.92i)3-s − 2.00·4-s + (−2.12 + 1.22i)5-s + (4.13 + 0.939i)6-s + (−6.56 − 2.44i)7-s + 2.82i·8-s + (−8.11 − 3.88i)9-s + (1.73 + 3.01i)10-s + (−9.75 − 5.63i)11-s + (1.32 − 5.85i)12-s + (−9.55 + 16.5i)13-s + (−3.45 + 9.27i)14-s + (−2.18 − 7.04i)15-s + 4.00·16-s + (8.32 − 4.80i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.221 + 0.975i)3-s − 0.500·4-s + (−0.425 + 0.245i)5-s + (0.689 + 0.156i)6-s + (−0.937 − 0.348i)7-s + 0.353i·8-s + (−0.901 − 0.432i)9-s + (0.173 + 0.301i)10-s + (−0.887 − 0.512i)11-s + (0.110 − 0.487i)12-s + (−0.735 + 1.27i)13-s + (−0.246 + 0.662i)14-s + (−0.145 − 0.469i)15-s + 0.250·16-s + (0.489 − 0.282i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.844 - 0.535i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.844 - 0.535i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0707830 + 0.243845i\)
\(L(\frac12)\) \(\approx\) \(0.0707830 + 0.243845i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (0.664 - 2.92i)T \)
7 \( 1 + (6.56 + 2.44i)T \)
good5 \( 1 + (2.12 - 1.22i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (9.75 + 5.63i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (9.55 - 16.5i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-8.32 + 4.80i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (8.82 - 15.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-28.0 + 16.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (25.7 - 14.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 19.3T + 961T^{2} \)
37 \( 1 + (15.3 - 26.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-10.3 - 5.99i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (30.0 + 52.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 41.8iT - 2.20e3T^{2} \)
53 \( 1 + (-20.2 + 11.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 4.86iT - 3.48e3T^{2} \)
61 \( 1 + 33.3T + 3.72e3T^{2} \)
67 \( 1 + 108.T + 4.48e3T^{2} \)
71 \( 1 - 133. iT - 5.04e3T^{2} \)
73 \( 1 + (-60.3 - 104. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 - 26.0T + 6.24e3T^{2} \)
83 \( 1 + (59.7 - 34.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (128. + 74.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-11.3 - 19.5i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.47962276484559372322802617952, −12.30703018927455622834036596760, −11.36428733632661244194800406883, −10.43486214058815190459391241611, −9.690776846446777396532813742035, −8.614390821085541007305580007966, −6.98589660471905147820544746419, −5.39541811558966403474909055841, −4.05757697937466512753031956634, −2.97253806207699742326795412608, 0.17075000006805915327693105284, 2.87552997812999374330481626095, 5.04896959986293860408433851479, 6.09471134241398490476577731607, 7.35369431437475185241602575319, 8.005799156933505249100237734590, 9.334778369322920839383488746109, 10.61536783630513707321017953272, 12.13180522114908710132749223481, 12.85178819282036102342506833330

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