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Properties

Label 2-126-63.20-c3-0-20
Degree $2$
Conductor $126$
Sign $-0.451 + 0.892i$
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 + (−2.71 + 4.42i)3-s + (1.99 + 3.46i)4-s + (−7.20 − 12.4i)5-s + (−9.13 + 4.95i)6-s + (−17.9 − 4.42i)7-s + 7.99i·8-s + (−12.2 − 24.0i)9-s − 28.8i·10-s + (−14.4 − 8.32i)11-s + (−20.7 − 0.550i)12-s + (22.2 − 12.8i)13-s + (−26.7 − 25.6i)14-s + (74.8 + 1.98i)15-s + (−8 + 13.8i)16-s − 95.0·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.522 + 0.852i)3-s + (0.249 + 0.433i)4-s + (−0.644 − 1.11i)5-s + (−0.621 + 0.337i)6-s + (−0.971 − 0.239i)7-s + 0.353i·8-s + (−0.453 − 0.891i)9-s − 0.911i·10-s + (−0.395 − 0.228i)11-s + (−0.499 − 0.0132i)12-s + (0.475 − 0.274i)13-s + (−0.510 − 0.489i)14-s + (1.28 + 0.0341i)15-s + (−0.125 + 0.216i)16-s − 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.451 + 0.892i$
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: \((2,\ 126,\ (\ :3/2),\ -0.451 + 0.892i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.175099 - 0.285011i\)
\(L(\frac12)\) \(\approx\) \(0.175099 - 0.285011i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.71 - 4.42i)T \)
7 \( 1 + (17.9 + 4.42i)T \)
good5 \( 1 + (7.20 + 12.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (14.4 + 8.32i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-22.2 + 12.8i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 95.0T + 4.91e3T^{2} \)
19 \( 1 - 29.3iT - 6.85e3T^{2} \)
23 \( 1 + (-82.8 + 47.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (189. + 109. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (176. - 102. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 317.T + 5.06e4T^{2} \)
41 \( 1 + (-182. - 316. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (189. - 328. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-128. + 222. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 610. iT - 1.48e5T^{2} \)
59 \( 1 + (-107. - 186. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-52.4 - 30.2i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-411. - 713. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 710. iT - 3.57e5T^{2} \)
73 \( 1 + 219. iT - 3.89e5T^{2} \)
79 \( 1 + (-56.3 + 97.6i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-550. + 953. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 190.T + 7.04e5T^{2} \)
97 \( 1 + (1.28e3 + 740. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.76423741716451868431544706182, −11.62241514031032492057647423334, −10.66867721943563004965151602214, −9.295127301237602540556631589077, −8.387093716390452512121627525438, −6.77395063971489647411169367472, −5.56097657724271555384140925182, −4.48261572796040555522844282055, −3.46611188638637838261256622504, −0.14212679764713724531920722500, 2.32525426767725454192781308527, 3.65311288918932549915339012116, 5.49869880540302187139345955323, 6.74886711842661429917910247424, 7.25083774192254779555077642031, 9.065693672864131911452147600174, 10.78478833287854498313718165674, 11.11720334705406223067665682240, 12.30932488502739241818219311003, 13.10564335135388216389817656727

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