Properties

Label 2-126-7.2-c11-0-27
Degree $2$
Conductor $126$
Sign $-0.448 + 0.893i$
Analytic cond. $96.8112$
Root an. cond. $9.83927$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16 − 27.7i)2-s + (−511. − 886. i)4-s + (−2.66e3 + 4.61e3i)5-s + (3.86e4 − 2.20e4i)7-s − 3.27e4·8-s + (8.53e4 + 1.47e5i)10-s + (−4.56e5 − 7.90e5i)11-s + 2.26e6·13-s + (6.34e3 − 1.42e6i)14-s + (−5.24e5 + 9.08e5i)16-s + (8.60e5 + 1.49e6i)17-s + (−4.81e6 + 8.34e6i)19-s + 5.46e6·20-s − 2.92e7·22-s + (1.05e6 − 1.82e6i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.381 + 0.661i)5-s + (0.868 − 0.496i)7-s − 0.353·8-s + (0.269 + 0.467i)10-s + (−0.854 − 1.48i)11-s + 1.69·13-s + (0.00315 − 0.707i)14-s + (−0.125 + 0.216i)16-s + (0.146 + 0.254i)17-s + (−0.446 + 0.772i)19-s + 0.381·20-s − 1.20·22-s + (0.0340 − 0.0590i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.448 + 0.893i$
Analytic conductor: \(96.8112\)
Root analytic conductor: \(9.83927\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :11/2),\ -0.448 + 0.893i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.359715102\)
\(L(\frac12)\) \(\approx\) \(2.359715102\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-16 + 27.7i)T \)
3 \( 1 \)
7 \( 1 + (-3.86e4 + 2.20e4i)T \)
good5 \( 1 + (2.66e3 - 4.61e3i)T + (-2.44e7 - 4.22e7i)T^{2} \)
11 \( 1 + (4.56e5 + 7.90e5i)T + (-1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 - 2.26e6T + 1.79e12T^{2} \)
17 \( 1 + (-8.60e5 - 1.49e6i)T + (-1.71e13 + 2.96e13i)T^{2} \)
19 \( 1 + (4.81e6 - 8.34e6i)T + (-5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (-1.05e6 + 1.82e6i)T + (-4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 - 7.26e6T + 1.22e16T^{2} \)
31 \( 1 + (-5.34e7 - 9.26e7i)T + (-1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (-4.25e7 + 7.36e7i)T + (-8.89e16 - 1.54e17i)T^{2} \)
41 \( 1 - 2.55e8T + 5.50e17T^{2} \)
43 \( 1 + 2.24e8T + 9.29e17T^{2} \)
47 \( 1 + (-9.77e8 + 1.69e9i)T + (-1.23e18 - 2.14e18i)T^{2} \)
53 \( 1 + (2.36e9 + 4.08e9i)T + (-4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (8.36e8 + 1.44e9i)T + (-1.50e19 + 2.61e19i)T^{2} \)
61 \( 1 + (-3.45e9 + 5.98e9i)T + (-2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (3.35e8 + 5.81e8i)T + (-6.10e19 + 1.05e20i)T^{2} \)
71 \( 1 + 3.52e9T + 2.31e20T^{2} \)
73 \( 1 + (1.51e10 + 2.61e10i)T + (-1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (1.52e10 - 2.63e10i)T + (-3.73e20 - 6.47e20i)T^{2} \)
83 \( 1 - 5.71e10T + 1.28e21T^{2} \)
89 \( 1 + (-1.76e10 + 3.05e10i)T + (-1.38e21 - 2.40e21i)T^{2} \)
97 \( 1 - 7.27e10T + 7.15e21T^{2} \)
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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

−10.89094518340431102795883430026, −10.48824931699688769762180159415, −8.648450262412941702032682429390, −7.948234772106253769837407048454, −6.40009273710291639250884495887, −5.34529949882994962305794127344, −3.87375293307602481724290742712, −3.17398432232229735276498030471, −1.62940378550897640922659460322, −0.51725142267839487640740760393, 1.09690556598252678359578913894, 2.51046807732585335291116242150, 4.24535682515632988624036258865, 4.90227188377117931046630097178, 6.06537707581048731671416032600, 7.45686780093982004988302767462, 8.296600741274980970331188097751, 9.143500285108434538010651326972, 10.67145223174822297815624630746, 11.77628668271874321290239163400

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