Properties

Label 2-126-63.20-c3-0-12
Degree $2$
Conductor $126$
Sign $-0.0903 - 0.995i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 + i)2-s + (4.36 + 2.81i)3-s + (1.99 + 3.46i)4-s + (5.62 + 9.74i)5-s + (4.75 + 9.24i)6-s + (−17.3 + 6.56i)7-s + 7.99i·8-s + (11.1 + 24.5i)9-s + 22.5i·10-s + (−52.7 − 30.4i)11-s + (−1.00 + 20.7i)12-s + (73.7 − 42.5i)13-s + (−36.5 − 5.95i)14-s + (−2.83 + 58.4i)15-s + (−8 + 13.8i)16-s + 54.9·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.840 + 0.541i)3-s + (0.249 + 0.433i)4-s + (0.503 + 0.871i)5-s + (0.323 + 0.628i)6-s + (−0.935 + 0.354i)7-s + 0.353i·8-s + (0.413 + 0.910i)9-s + 0.711i·10-s + (−1.44 − 0.834i)11-s + (−0.0242 + 0.499i)12-s + (1.57 − 0.907i)13-s + (−0.697 − 0.113i)14-s + (−0.0488 + 1.00i)15-s + (−0.125 + 0.216i)16-s + 0.784·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0903 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0903 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.0903 - 0.995i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ -0.0903 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.94347 + 2.12770i\)
\(L(\frac12)\) \(\approx\) \(1.94347 + 2.12770i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 - i)T \)
3 \( 1 + (-4.36 - 2.81i)T \)
7 \( 1 + (17.3 - 6.56i)T \)
good5 \( 1 + (-5.62 - 9.74i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (52.7 + 30.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-73.7 + 42.5i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 54.9T + 4.91e3T^{2} \)
19 \( 1 - 9.90iT - 6.85e3T^{2} \)
23 \( 1 + (-6.77 + 3.91i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-204. - 117. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-53.1 + 30.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 218.T + 5.06e4T^{2} \)
41 \( 1 + (-36.9 - 63.9i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-116. + 201. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-43.8 + 76.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 56.2iT - 1.48e5T^{2} \)
59 \( 1 + (-217. - 376. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (715. + 413. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (451. + 782. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 894. iT - 3.57e5T^{2} \)
73 \( 1 + 629. iT - 3.89e5T^{2} \)
79 \( 1 + (469. - 812. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (236. - 409. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 417.T + 7.04e5T^{2} \)
97 \( 1 + (485. + 280. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.55605871185507243947930410303, −12.56102144930109685418103029703, −10.68177329432241461624590242522, −10.33421849341927278887979714412, −8.776272348302010430905947171554, −7.84415859831260357333881222880, −6.34364597297115242376631635487, −5.39290652286005596300672559477, −3.37808493636668900622230786269, −2.84623247353760332276223438820, 1.30631898179591294259270842614, 2.86311973287199477196284013608, 4.29091094161631363276908771444, 5.85340117601856134266401283908, 7.05694718664538135695524302789, 8.430290719899369384290634055777, 9.514811384314318750124075641360, 10.40426134932358077889771151916, 12.06480643694959596290350937920, 12.97227528672961237441688263584

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