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Properties

Label 2-126-21.11-c8-0-14
Degree $2$
Conductor $126$
Sign $0.656 + 0.754i$
Analytic cond. $51.3297$
Root an. cond. $7.16447$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.79 − 5.65i)2-s + (63.9 − 110. i)4-s + (675. − 389. i)5-s + (−2.25e3 + 818. i)7-s − 1.44e3i·8-s + (4.41e3 − 7.63e3i)10-s + (2.23e4 + 1.29e4i)11-s + 2.64e3·13-s + (−1.74e4 + 2.07e4i)14-s + (−8.19e3 − 1.41e4i)16-s + (6.41e4 + 3.70e4i)17-s + (−1.99e4 − 3.45e4i)19-s − 9.98e4i·20-s + 2.91e5·22-s + (3.91e5 − 2.25e5i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (1.08 − 0.623i)5-s + (−0.940 + 0.340i)7-s − 0.353i·8-s + (0.441 − 0.763i)10-s + (1.52 + 0.881i)11-s + 0.0926·13-s + (−0.455 + 0.541i)14-s + (−0.125 − 0.216i)16-s + (0.767 + 0.443i)17-s + (−0.153 − 0.265i)19-s − 0.623i·20-s + 1.24·22-s + (1.39 − 0.806i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.656 + 0.754i$
Analytic conductor: \(51.3297\)
Root analytic conductor: \(7.16447\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :4),\ 0.656 + 0.754i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.884284577\)
\(L(\frac12)\) \(\approx\) \(3.884284577\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.79 + 5.65i)T \)
3 \( 1 \)
7 \( 1 + (2.25e3 - 818. i)T \)
good5 \( 1 + (-675. + 389. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-2.23e4 - 1.29e4i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 2.64e3T + 8.15e8T^{2} \)
17 \( 1 + (-6.41e4 - 3.70e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (1.99e4 + 3.45e4i)T + (-8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-3.91e5 + 2.25e5i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 1.61e4iT - 5.00e11T^{2} \)
31 \( 1 + (1.30e5 - 2.26e5i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (7.04e5 + 1.21e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 - 1.52e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.63e6T + 1.16e13T^{2} \)
47 \( 1 + (-8.03e6 + 4.63e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-2.72e6 - 1.57e6i)T + (3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (1.44e7 + 8.35e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-4.37e6 - 7.56e6i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-1.40e7 + 2.43e7i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 4.04e7iT - 6.45e14T^{2} \)
73 \( 1 + (-5.58e6 + 9.68e6i)T + (-4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-3.35e7 - 5.81e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + 4.64e7iT - 2.25e15T^{2} \)
89 \( 1 + (3.34e7 - 1.93e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 - 9.56e7T + 7.83e15T^{2} \)
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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

−12.09408831882124396290039775765, −10.57724490557792668538719618765, −9.518097953347917250637581466549, −8.991282706311559922360416904603, −6.89393455995229230642224668885, −6.05448945149440853064025209668, −4.92487013744180453593220890237, −3.58235041694656113757021048733, −2.13260840437589707634279494951, −1.01757367099571373747658247101, 1.15663171374428917328725981391, 2.87475301718055569229848336416, 3.79287784670941844381487749613, 5.58536461732714023468294416224, 6.38051354824377111294520280281, 7.18240347621554927057483717924, 8.944317416315779627628885070281, 9.817649626974050530682276518719, 10.97345107353065900385121423930, 12.08893988088332666111727610806

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