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Properties

Label 2-126-63.20-c3-0-19
Degree $2$
Conductor $126$
Sign $0.896 + 0.443i$
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 + (4.77 − 2.05i)3-s + (1.99 + 3.46i)4-s + (−9.07 − 15.7i)5-s + (10.3 + 1.20i)6-s + (15.5 + 10.0i)7-s + 7.99i·8-s + (18.5 − 19.6i)9-s − 36.3i·10-s + (−3.14 − 1.81i)11-s + (16.6 + 12.4i)12-s + (78.5 − 45.3i)13-s + (16.8 + 33.0i)14-s + (−75.6 − 56.3i)15-s + (−8 + 13.8i)16-s − 36.8·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.918 − 0.396i)3-s + (0.249 + 0.433i)4-s + (−0.811 − 1.40i)5-s + (0.702 + 0.0820i)6-s + (0.838 + 0.544i)7-s + 0.353i·8-s + (0.686 − 0.727i)9-s − 1.14i·10-s + (−0.0860 − 0.0496i)11-s + (0.401 + 0.298i)12-s + (1.67 − 0.967i)13-s + (0.320 + 0.630i)14-s + (−1.30 − 0.969i)15-s + (−0.125 + 0.216i)16-s − 0.525·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.896 + 0.443i$
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: \((2,\ 126,\ (\ :3/2),\ 0.896 + 0.443i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.84557 - 0.666136i\)
\(L(\frac12)\) \(\approx\) \(2.84557 - 0.666136i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.77 + 2.05i)T \)
7 \( 1 + (-15.5 - 10.0i)T \)
good5 \( 1 + (9.07 + 15.7i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (3.14 + 1.81i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-78.5 + 45.3i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 36.8T + 4.91e3T^{2} \)
19 \( 1 - 15.7iT - 6.85e3T^{2} \)
23 \( 1 + (145. - 83.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-129. - 74.7i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (230. - 132. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 249.T + 5.06e4T^{2} \)
41 \( 1 + (-11.4 - 19.8i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (133. - 231. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (176. - 306. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 202. iT - 1.48e5T^{2} \)
59 \( 1 + (261. + 452. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (311. + 179. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-252. - 437. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 232. iT - 3.57e5T^{2} \)
73 \( 1 + 899. iT - 3.89e5T^{2} \)
79 \( 1 + (63.9 - 110. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (254. - 440. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.25e3T + 7.04e5T^{2} \)
97 \( 1 + (-724. - 418. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.89400292475837926469652430392, −12.20631319374819717412592477334, −11.11342791838679989529219428555, −9.108484956125753448497247394658, −8.247920451314369090465703376901, −7.85390050015074013040013094132, −5.98510404028676144367748469083, −4.65941951190786562247239093831, −3.49994872662435828585624373474, −1.43581695088963489483864771685, 2.14805510786045111372928579755, 3.67808387490435602085935334044, 4.29024447606486741057035344511, 6.44357403386800341116865021581, 7.56237332776027659102017655984, 8.612264358575245391351444755953, 10.21813518788323828885733866068, 11.00155639119832823047219903745, 11.62607653758542249841579240675, 13.37549158678232768635248937246

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