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Properties

Label 2-126-9.5-c2-0-7
Degree $2$
Conductor $126$
Sign $0.873 + 0.487i$
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 + (2.99 + 0.240i)3-s + (0.999 − 1.73i)4-s + (0.665 + 0.384i)5-s + (3.83 − 1.82i)6-s + (−1.32 − 2.29i)7-s − 2.82i·8-s + (8.88 + 1.43i)9-s + 1.08·10-s + (−3.15 + 1.82i)11-s + (3.40 − 4.93i)12-s + (−2.18 + 3.77i)13-s + (−3.24 − 1.87i)14-s + (1.89 + 1.30i)15-s + (−2.00 − 3.46i)16-s + 8.55i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.996 + 0.0800i)3-s + (0.249 − 0.433i)4-s + (0.133 + 0.0768i)5-s + (0.638 − 0.303i)6-s + (−0.188 − 0.327i)7-s − 0.353i·8-s + (0.987 + 0.159i)9-s + 0.108·10-s + (−0.286 + 0.165i)11-s + (0.283 − 0.411i)12-s + (−0.167 + 0.290i)13-s + (−0.231 − 0.133i)14-s + (0.126 + 0.0872i)15-s + (−0.125 − 0.216i)16-s + 0.503i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.873 + 0.487i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.873 + 0.487i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.43301 - 0.633399i\)
\(L(\frac12)\) \(\approx\) \(2.43301 - 0.633399i\)
\(L(2)\) 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 \( 1 + (-2.99 - 0.240i)T \)
7 \( 1 + (1.32 + 2.29i)T \)
good5 \( 1 + (-0.665 - 0.384i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (3.15 - 1.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (2.18 - 3.77i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 8.55iT - 289T^{2} \)
19 \( 1 + 18.7T + 361T^{2} \)
23 \( 1 + (-10.8 - 6.26i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (14.2 - 8.21i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (6.88 - 11.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 28.3T + 1.36e3T^{2} \)
41 \( 1 + (-51.3 - 29.6i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (25.7 + 44.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (48.0 - 27.7i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 51.2iT - 2.80e3T^{2} \)
59 \( 1 + (-55.7 - 32.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (24.6 + 42.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-24.6 + 42.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 81.1iT - 5.04e3T^{2} \)
73 \( 1 - 140.T + 5.32e3T^{2} \)
79 \( 1 + (69.7 + 120. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-116. + 67.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 8.94iT - 7.92e3T^{2} \)
97 \( 1 + (-45.2 - 78.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.13343999393897601275097185953, −12.37109689971589359496647766233, −10.87006338150758691382109084726, −10.02456633906276210820186627860, −8.914737856737883381741207817977, −7.63469459066333433431702142675, −6.41184950643219551199678209355, −4.68487846752349710613755064180, −3.50043629669992014570836761126, −2.04086027698174369839903178900, 2.37861755344994660352101842368, 3.73953766726472106729415183738, 5.22528931162062086147335120167, 6.67761322333840869295829262261, 7.80825180991220408878011582693, 8.819266892832675505864968266389, 9.903366196040260423135435915024, 11.31165303463920007953020474317, 12.71486845818693440139827635765, 13.18696796886917857867991248357

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