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Properties

Label 2-126-9.2-c2-0-11
Degree $2$
Conductor $126$
Sign $0.912 + 0.408i$
Analytic cond. $3.43325$
Root an. cond. $1.85290$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (1.12 − 2.78i)3-s + (0.999 + 1.73i)4-s + (2.56 − 1.47i)5-s + (3.34 − 2.60i)6-s + (1.32 − 2.29i)7-s + 2.82i·8-s + (−6.45 − 6.26i)9-s + 4.18·10-s + (1.94 + 1.12i)11-s + (5.94 − 0.828i)12-s + (2.24 + 3.88i)13-s + (3.24 − 1.87i)14-s + (−1.22 − 8.78i)15-s + (−2.00 + 3.46i)16-s + 3.17i·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.375 − 0.926i)3-s + (0.249 + 0.433i)4-s + (0.512 − 0.295i)5-s + (0.557 − 0.434i)6-s + (0.188 − 0.327i)7-s + 0.353i·8-s + (−0.717 − 0.696i)9-s + 0.418·10-s + (0.176 + 0.101i)11-s + (0.495 − 0.0690i)12-s + (0.172 + 0.298i)13-s + (0.231 − 0.133i)14-s + (−0.0816 − 0.585i)15-s + (−0.125 + 0.216i)16-s + 0.186i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.912 + 0.408i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.912 + 0.408i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.21799 - 0.474109i\)
\(L(\frac12)\) \(\approx\) \(2.21799 - 0.474109i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (-1.12 + 2.78i)T \)
7 \( 1 + (-1.32 + 2.29i)T \)
good5 \( 1 + (-2.56 + 1.47i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-1.94 - 1.12i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-2.24 - 3.88i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 3.17iT - 289T^{2} \)
19 \( 1 - 15.1T + 361T^{2} \)
23 \( 1 + (18.1 - 10.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (28.8 + 16.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-25.7 - 44.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 8.74T + 1.36e3T^{2} \)
41 \( 1 + (61.1 - 35.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-22.4 + 38.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (16.1 + 9.34i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 38.6iT - 2.80e3T^{2} \)
59 \( 1 + (33.3 - 19.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-36.6 + 63.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-54.4 - 94.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 23.2iT - 5.04e3T^{2} \)
73 \( 1 - 47.8T + 5.32e3T^{2} \)
79 \( 1 + (-39.2 + 67.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-98.6 - 56.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 164. iT - 7.92e3T^{2} \)
97 \( 1 + (-43.8 + 75.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.37870407322900871925264818084, −12.25668305567607560342817435305, −11.43106058411673597882199008142, −9.775636381831400666255145610289, −8.549794264205212969581366991723, −7.49036302884620458559507168311, −6.45933205495015326821067735959, −5.29127347431254637188895418702, −3.54626925394930200604766187374, −1.73712110134854628424020831567, 2.39194091193521222598336183108, 3.74653835602756910990272424122, 5.11189396521704678817466241188, 6.16548507530998801413115563365, 7.960978739896984715296902676231, 9.307561882401244847523622771358, 10.13093388576375128852165399444, 11.10578978296216864053806031493, 12.08885591325786648276400951691, 13.49816467629431002151225211520

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