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Properties

Label 2-126-63.41-c3-0-11
Degree $2$
Conductor $126$
Sign $0.840 + 0.542i$
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 + (5.04 − 1.25i)3-s + (1.99 − 3.46i)4-s + (−0.984 + 1.70i)5-s + (−7.47 + 7.21i)6-s + (3.48 − 18.1i)7-s + 7.99i·8-s + (23.8 − 12.6i)9-s − 3.93i·10-s + (15.6 − 9.06i)11-s + (5.72 − 19.9i)12-s + (−7.54 − 4.35i)13-s + (12.1 + 34.9i)14-s + (−2.81 + 9.83i)15-s + (−8 − 13.8i)16-s − 24.9·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.970 − 0.242i)3-s + (0.249 − 0.433i)4-s + (−0.0880 + 0.152i)5-s + (−0.508 + 0.491i)6-s + (0.188 − 0.982i)7-s + 0.353i·8-s + (0.882 − 0.469i)9-s − 0.124i·10-s + (0.430 − 0.248i)11-s + (0.137 − 0.480i)12-s + (−0.161 − 0.0929i)13-s + (0.231 + 0.667i)14-s + (−0.0485 + 0.169i)15-s + (−0.125 − 0.216i)16-s − 0.356·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.840 + 0.542i$
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: \((2,\ 126,\ (\ :3/2),\ 0.840 + 0.542i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.62630 - 0.479244i\)
\(L(\frac12)\) \(\approx\) \(1.62630 - 0.479244i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.04 + 1.25i)T \)
7 \( 1 + (-3.48 + 18.1i)T \)
good5 \( 1 + (0.984 - 1.70i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-15.6 + 9.06i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (7.54 + 4.35i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 24.9T + 4.91e3T^{2} \)
19 \( 1 + 147. iT - 6.85e3T^{2} \)
23 \( 1 + (-133. - 77.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-171. + 99.2i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-18.9 - 10.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 4.61T + 5.06e4T^{2} \)
41 \( 1 + (209. - 363. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (161. + 279. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (102. + 176. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 512. iT - 1.48e5T^{2} \)
59 \( 1 + (296. - 513. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (294. - 169. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (268. - 465. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 13.3iT - 3.57e5T^{2} \)
73 \( 1 - 27.3iT - 3.89e5T^{2} \)
79 \( 1 + (-230. - 399. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (43.2 + 74.8i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 336.T + 7.04e5T^{2} \)
97 \( 1 + (155. - 89.6i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.24174414208873449989189480775, −11.57515126356145114246850729947, −10.54408436812971615539624206421, −9.394560795175820752688770585965, −8.565216122528509063655662332926, −7.34460442470907078229279735272, −6.75467819068892457404300951545, −4.64629640452951565109557379720, −3.01042949176079002393575535925, −1.09441313401117846319706639598, 1.79176636190281099505467955491, 3.15926322315658829477984811117, 4.72415020449087386724728365662, 6.63056405075632324272009660845, 8.097878032487534407426154964589, 8.734373441270569046043086152515, 9.682972419678639094834834889696, 10.68987301304796776124237877805, 12.09608718947950038781948115134, 12.72147559625949467376487784055

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