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Properties

Label 2-126-63.13-c2-0-9
Degree $2$
Conductor $126$
Sign $0.945 + 0.326i$
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 + (2.95 − 0.522i)3-s + (−0.999 + 1.73i)4-s + (4.10 + 2.36i)5-s + (−2.72 − 3.24i)6-s + (1.32 + 6.87i)7-s + 2.82·8-s + (8.45 − 3.08i)9-s − 6.70i·10-s + (3.85 + 6.67i)11-s + (−2.04 + 5.63i)12-s + (−13.7 − 7.91i)13-s + (7.48 − 6.47i)14-s + (13.3 + 4.85i)15-s + (−2.00 − 3.46i)16-s − 4.53i·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.984 − 0.174i)3-s + (−0.249 + 0.433i)4-s + (0.820 + 0.473i)5-s + (−0.454 − 0.541i)6-s + (0.188 + 0.982i)7-s + 0.353·8-s + (0.939 − 0.343i)9-s − 0.670i·10-s + (0.350 + 0.607i)11-s + (−0.170 + 0.469i)12-s + (−1.05 − 0.608i)13-s + (0.534 − 0.462i)14-s + (0.890 + 0.323i)15-s + (−0.125 − 0.216i)16-s − 0.266i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.945 + 0.326i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.72358 - 0.289368i\)
\(L(\frac12)\) \(\approx\) \(1.72358 - 0.289368i\)
\(L(2)\) 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 \( 1 + (-2.95 + 0.522i)T \)
7 \( 1 + (-1.32 - 6.87i)T \)
good5 \( 1 + (-4.10 - 2.36i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.85 - 6.67i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (13.7 + 7.91i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 4.53iT - 289T^{2} \)
19 \( 1 + 17.9iT - 361T^{2} \)
23 \( 1 + (-7.38 + 12.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-12.9 - 22.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (35.5 + 20.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 55.1T + 1.36e3T^{2} \)
41 \( 1 + (-20.3 - 11.7i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-31.8 - 55.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (20.3 - 11.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 41.2T + 2.80e3T^{2} \)
59 \( 1 + (69.7 + 40.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-4.59 + 2.65i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-55.9 + 96.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 90.1T + 5.04e3T^{2} \)
73 \( 1 - 108. iT - 5.32e3T^{2} \)
79 \( 1 + (-0.901 - 1.56i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-133. + 77.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 114. iT - 7.92e3T^{2} \)
97 \( 1 + (132. - 76.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.88634078443131470663893092761, −12.29368623858755444176252185632, −10.84868292752226409493482806501, −9.653792532230274943236713358213, −9.185265479494418278417422834834, −7.905620012044522951104453919059, −6.70318033030670427012690933983, −4.90466816212355294898671263106, −2.93668617454876798472595795282, −2.05165882546509595478461828952, 1.67169095475846435181889088755, 3.87365469987305487567095986488, 5.27899360049058487963176308136, 6.86960474797195627656615790152, 7.85869498091760052439678003494, 9.002196284963740595713520546296, 9.737114758343571980862789674197, 10.68521193812243641649295254759, 12.45887775840818787944072421101, 13.76476104774292659410960154153

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