Properties

Degree $2$
Conductor $126$
Sign $-0.0165 + 0.999i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (3.5 + 6.06i)5-s + (−10 − 15.5i)7-s + 7.99·8-s + (7 − 12.1i)10-s + (17.5 − 30.3i)11-s + 66·13-s + (−17 + 32.9i)14-s + (−8 − 13.8i)16-s + (29.5 − 51.0i)17-s + (−68.5 − 118. i)19-s − 28·20-s − 70·22-s + (−3.5 − 6.06i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.313 + 0.542i)5-s + (−0.539 − 0.841i)7-s + 0.353·8-s + (0.221 − 0.383i)10-s + (0.479 − 0.830i)11-s + 1.40·13-s + (−0.324 + 0.628i)14-s + (−0.125 − 0.216i)16-s + (0.420 − 0.728i)17-s + (−0.827 − 1.43i)19-s − 0.313·20-s − 0.678·22-s + (−0.0317 − 0.0549i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0165 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.0165 + 0.999i$
Motivic weight: \(3\)
Character: $\chi_{126} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ -0.0165 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.895753 - 0.910745i\)
\(L(\frac12)\) \(\approx\) \(0.895753 - 0.910745i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 \)
7 \( 1 + (10 + 15.5i)T \)
good5 \( 1 + (-3.5 - 6.06i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-17.5 + 30.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 66T + 2.19e3T^{2} \)
17 \( 1 + (-29.5 + 51.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (68.5 + 118. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 106T + 2.43e4T^{2} \)
31 \( 1 + (37.5 - 64.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 498T + 6.89e4T^{2} \)
43 \( 1 - 260T + 7.95e4T^{2} \)
47 \( 1 + (85.5 + 148. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (208.5 - 361. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (8.5 - 14.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (25.5 + 44.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (219.5 - 380. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 784T + 3.57e5T^{2} \)
73 \( 1 + (147.5 - 255. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-247.5 - 428. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 932T + 5.71e5T^{2} \)
89 \( 1 + (436.5 + 756. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 290T + 9.12e5T^{2} \)
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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

−12.70553890088900396898938867677, −11.12775368802122536829384933195, −10.85187120397412195963459934162, −9.556365322412541724954243749666, −8.623291758005816389457599828309, −7.15353843643374221300749993991, −6.10482947107831492140650999615, −4.09579777439409544376853562739, −2.89321930069236891371525920164, −0.819899118390465931730992080482, 1.62684346070702057831138793024, 3.94231990908140349888808183708, 5.65129927973124152435328074401, 6.34742217827225993570876364248, 7.929807785149124856465909414991, 8.937302555350555764990247611080, 9.686843395770688861954252744410, 10.94472754849607374695853521472, 12.43839036002574441614819659114, 13.00579591233581098222925099371

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