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Properties

Label 2-126-7.4-c11-0-31
Degree $2$
Conductor $126$
Sign $0.992 + 0.119i$
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 + (1.09e3 + 1.90e3i)5-s + (3.97e4 − 2.00e4i)7-s − 3.27e4·8-s + (−3.51e4 + 6.08e4i)10-s + (2.06e5 − 3.57e5i)11-s + 1.89e6·13-s + (1.19e6 + 7.79e5i)14-s + (−5.24e5 − 9.08e5i)16-s + (−4.55e6 + 7.88e6i)17-s + (−1.24e6 − 2.15e6i)19-s − 2.24e6·20-s + 1.32e7·22-s + (−2.95e7 − 5.11e7i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.157 + 0.272i)5-s + (0.892 − 0.450i)7-s − 0.353·8-s + (−0.111 + 0.192i)10-s + (0.386 − 0.669i)11-s + 1.41·13-s + (0.591 + 0.387i)14-s + (−0.125 − 0.216i)16-s + (−0.777 + 1.34i)17-s + (−0.115 − 0.199i)19-s − 0.157·20-s + 0.546·22-s + (−0.957 − 1.65i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.119i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.992 + 0.119i$
Analytic conductor: \(96.8112\)
Root analytic conductor: \(9.83927\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :11/2),\ 0.992 + 0.119i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.964381060\)
\(L(\frac12)\) \(\approx\) \(2.964381060\)
\(L(\frac{13}{2})\) 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 \( 1 + (-3.97e4 + 2.00e4i)T \)
good5 \( 1 + (-1.09e3 - 1.90e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-2.06e5 + 3.57e5i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 - 1.89e6T + 1.79e12T^{2} \)
17 \( 1 + (4.55e6 - 7.88e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (1.24e6 + 2.15e6i)T + (-5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (2.95e7 + 5.11e7i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 - 7.92e7T + 1.22e16T^{2} \)
31 \( 1 + (-5.46e7 + 9.46e7i)T + (-1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (3.38e8 + 5.85e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 + 2.52e8T + 5.50e17T^{2} \)
43 \( 1 + 1.75e9T + 9.29e17T^{2} \)
47 \( 1 + (-1.29e9 - 2.23e9i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (-2.44e9 + 4.23e9i)T + (-4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (1.46e9 - 2.53e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (5.72e8 + 9.92e8i)T + (-2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (-5.82e9 + 1.00e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 + 5.24e9T + 2.31e20T^{2} \)
73 \( 1 + (-8.75e9 + 1.51e10i)T + (-1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (-2.12e10 - 3.67e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 8.46e9T + 1.28e21T^{2} \)
89 \( 1 + (9.36e9 + 1.62e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 - 1.03e11T + 7.15e21T^{2} \)
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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

−11.10624215654867304991535880022, −10.45123693198746189625826735394, −8.582368168015786127136005733590, −8.277929180335006627563420922855, −6.65055064588992988445902497079, −6.03709150896260449397271881702, −4.52599971386766965079465445872, −3.68958251172123835977012851837, −2.02636455583095651333419134114, −0.61667156089895694408651362413, 1.16093136677184723105317584364, 1.92314788838172332360216880237, 3.38372441779726583352840305050, 4.63466096381608203794954813648, 5.50956491482794122332395296166, 6.87342808031165125866472835029, 8.375926275265192468672195556995, 9.209854432015167111167346076301, 10.36970713549414135541075159508, 11.58485386499962716189478957391

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