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Properties

Label 2-126-21.11-c6-0-10
Degree $2$
Conductor $126$
Sign $0.999 + 0.0403i$
Analytic cond. $28.9868$
Root an. cond. $5.38394$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.89 + 2.82i)2-s + (15.9 − 27.7i)4-s + (201. − 116. i)5-s + (342. + 23.5i)7-s + 181. i·8-s + (−658. + 1.14e3i)10-s + (1.60e3 + 924. i)11-s + 2.87e3·13-s + (−1.74e3 + 852. i)14-s + (−512. − 886. i)16-s + (−1.75e3 − 1.01e3i)17-s + (−1.91e3 − 3.31e3i)19-s − 7.44e3i·20-s − 1.04e4·22-s + (−1.81e4 + 1.04e4i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (1.61 − 0.931i)5-s + (0.997 + 0.0687i)7-s + 0.353i·8-s + (−0.658 + 1.14i)10-s + (1.20 + 0.694i)11-s + 1.31·13-s + (−0.635 + 0.310i)14-s + (−0.125 − 0.216i)16-s + (−0.357 − 0.206i)17-s + (−0.279 − 0.483i)19-s − 0.931i·20-s − 0.982·22-s + (−1.49 + 0.862i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0403i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.999 + 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.999 + 0.0403i$
Analytic conductor: \(28.9868\)
Root analytic conductor: \(5.38394\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3),\ 0.999 + 0.0403i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.487795193\)
\(L(\frac12)\) \(\approx\) \(2.487795193\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.89 - 2.82i)T \)
3 \( 1 \)
7 \( 1 + (-342. - 23.5i)T \)
good5 \( 1 + (-201. + 116. i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-1.60e3 - 924. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 2.87e3T + 4.82e6T^{2} \)
17 \( 1 + (1.75e3 + 1.01e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (1.91e3 + 3.31e3i)T + (-2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (1.81e4 - 1.04e4i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 2.37e4iT - 5.94e8T^{2} \)
31 \( 1 + (-8.30e3 + 1.43e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-1.34e4 - 2.32e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 2.88e4iT - 4.75e9T^{2} \)
43 \( 1 + 5.04e4T + 6.32e9T^{2} \)
47 \( 1 + (2.95e4 - 1.70e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-1.79e5 - 1.03e5i)T + (1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (2.46e5 + 1.42e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.15e5 + 2.00e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (7.88e4 - 1.36e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 4.02e5iT - 1.28e11T^{2} \)
73 \( 1 + (-3.45e5 + 5.98e5i)T + (-7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-4.04e5 - 7.00e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 7.12e5iT - 3.26e11T^{2} \)
89 \( 1 + (7.56e4 - 4.36e4i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 6.34e5T + 8.32e11T^{2} \)
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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

−12.17142632958554582748885354293, −11.03214265534974150870964664008, −9.792640262564462687662988536979, −9.072444618688347690552342827741, −8.239286731461543464661754934096, −6.59221148917464240447209012401, −5.68048241832162393035973215928, −4.48541028136566650932161079006, −1.86565180933413741938339517328, −1.25642834170835322579583848557, 1.30060652264320597819745046103, 2.22067925307658164496163947541, 3.85475195556644672485163362370, 5.91657830170219151288396668928, 6.56120589378173023090526362121, 8.227297504791101428659143251837, 9.132743487803661811735592315570, 10.30933794882540873511068029198, 10.92030631380221013846671066920, 11.91609707023757525446089778993

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