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Properties

Label 2-126-63.41-c3-0-9
Degree $2$
Conductor $126$
Sign $0.275 - 0.961i$
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 + (2.30 + 4.65i)3-s + (1.99 − 3.46i)4-s + (2.59 − 4.50i)5-s + (−8.64 − 5.76i)6-s + (17.5 + 5.89i)7-s + 7.99i·8-s + (−16.4 + 21.4i)9-s + 10.3i·10-s + (60.1 − 34.7i)11-s + (20.7 + 1.34i)12-s + (21.9 + 12.6i)13-s + (−36.3 + 7.33i)14-s + (26.9 + 1.74i)15-s + (−8 − 13.8i)16-s − 102.·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.442 + 0.896i)3-s + (0.249 − 0.433i)4-s + (0.232 − 0.402i)5-s + (−0.588 − 0.392i)6-s + (0.947 + 0.318i)7-s + 0.353i·8-s + (−0.607 + 0.794i)9-s + 0.328i·10-s + (1.64 − 0.951i)11-s + (0.498 + 0.0323i)12-s + (0.468 + 0.270i)13-s + (−0.693 + 0.140i)14-s + (0.463 + 0.0301i)15-s + (−0.125 − 0.216i)16-s − 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.275 - 0.961i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.31020 + 0.987232i\)
\(L(\frac12)\) \(\approx\) \(1.31020 + 0.987232i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.30 - 4.65i)T \)
7 \( 1 + (-17.5 - 5.89i)T \)
good5 \( 1 + (-2.59 + 4.50i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-60.1 + 34.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-21.9 - 12.6i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 102.T + 4.91e3T^{2} \)
19 \( 1 - 118. iT - 6.85e3T^{2} \)
23 \( 1 + (-62.2 - 35.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-49.6 + 28.6i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (50.1 + 28.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 40.2T + 5.06e4T^{2} \)
41 \( 1 + (120. - 208. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-80.2 - 138. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (280. + 486. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 768. iT - 1.48e5T^{2} \)
59 \( 1 + (-86.9 + 150. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-357. + 206. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-271. + 469. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 870. iT - 3.57e5T^{2} \)
73 \( 1 + 822. iT - 3.89e5T^{2} \)
79 \( 1 + (-46.1 - 79.9i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (223. + 387. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 103.T + 7.04e5T^{2} \)
97 \( 1 + (1.26e3 - 729. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.44375970726245973442270185748, −11.57263212375325384798397905404, −11.07225052324961056425380872361, −9.646290361864313549338818256895, −8.792403851419957809886336256279, −8.280366976359009288758989377125, −6.46902441867983161911699061507, −5.18943223907623058942775866872, −3.81777226064544979033766461261, −1.64488462690483392046451880829, 1.19239671297041675739638366665, 2.46749968096759879650661670105, 4.31503882456554359701151358303, 6.60250328268064501627323112228, 7.16846656510747589755026741872, 8.593272664082622118170796866367, 9.225408865468046925343844128011, 10.80163998000046435898975348027, 11.55465344349645572323867532969, 12.59251917391991613598856835455

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