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Properties

Label 2-126-63.2-c2-0-14
Degree $2$
Conductor $126$
Sign $0.484 + 0.874i$
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  + (1.22 − 0.707i)2-s + (2.84 − 0.954i)3-s + (0.999 − 1.73i)4-s − 2.12i·5-s + (2.80 − 3.18i)6-s + (−5.95 − 3.68i)7-s − 2.82i·8-s + (7.17 − 5.43i)9-s + (−1.50 − 2.60i)10-s + 8.95i·11-s + (1.19 − 5.88i)12-s + (10.4 + 18.0i)13-s + (−9.89 − 0.305i)14-s + (−2.03 − 6.05i)15-s + (−2.00 − 3.46i)16-s + (−9.01 + 5.20i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.948 − 0.318i)3-s + (0.249 − 0.433i)4-s − 0.425i·5-s + (0.468 − 0.530i)6-s + (−0.850 − 0.526i)7-s − 0.353i·8-s + (0.797 − 0.603i)9-s + (−0.150 − 0.260i)10-s + 0.814i·11-s + (0.0992 − 0.490i)12-s + (0.801 + 1.38i)13-s + (−0.706 − 0.0218i)14-s + (−0.135 − 0.403i)15-s + (−0.125 − 0.216i)16-s + (−0.530 + 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.484 + 0.874i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.484 + 0.874i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.06709 - 1.21824i\)
\(L(\frac12)\) \(\approx\) \(2.06709 - 1.21824i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (-2.84 + 0.954i)T \)
7 \( 1 + (5.95 + 3.68i)T \)
good5 \( 1 + 2.12iT - 25T^{2} \)
11 \( 1 - 8.95iT - 121T^{2} \)
13 \( 1 + (-10.4 - 18.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (9.01 - 5.20i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.15 + 10.6i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 - 19.1iT - 529T^{2} \)
29 \( 1 + (28.3 + 16.3i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (18.9 - 32.8i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-16.9 + 29.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (28.7 - 16.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-10.3 + 17.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (57.3 - 33.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (2.72 - 1.57i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (72.9 + 42.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-46.0 - 79.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-33.3 + 57.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 115. iT - 5.04e3T^{2} \)
73 \( 1 + (19.6 + 33.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (26.2 + 45.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-9.25 - 5.34i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-61.9 - 35.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-64.5 + 111. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.14479996797346347097866883480, −12.28887116359693337829698354052, −11.00819974111341629291378045633, −9.615918676051241730278800243691, −8.989539867153572362143316874757, −7.33707138242060303920635269882, −6.45780582636273019756826088371, −4.51048394633166130167201529779, −3.44707027080103370919121036717, −1.72113777352975284866981362376, 2.81949366285120572609657062872, 3.64423287998879716248703739361, 5.45368593635453056520136525605, 6.63617926945986927405267477547, 7.999596817957746973698600466556, 8.895256232533217564123357309001, 10.16346097888099306125947252475, 11.21986331778890679239891134698, 12.83507771450016681623076840479, 13.26099099365160308180943089906

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