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Properties

Label 2-126-63.23-c2-0-4
Degree $2$
Conductor $126$
Sign $0.488 - 0.872i$
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  − 1.41i·2-s + (2.02 + 2.21i)3-s − 2.00·4-s + (−7.20 + 4.15i)5-s + (3.13 − 2.86i)6-s + (5.54 + 4.27i)7-s + 2.82i·8-s + (−0.819 + 8.96i)9-s + (5.88 + 10.1i)10-s + (9.13 + 5.27i)11-s + (−4.04 − 4.43i)12-s + (1.24 − 2.14i)13-s + (6.05 − 7.83i)14-s + (−23.7 − 7.55i)15-s + 4.00·16-s + (−27.5 + 15.8i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.674 + 0.738i)3-s − 0.500·4-s + (−1.44 + 0.831i)5-s + (0.522 − 0.476i)6-s + (0.791 + 0.611i)7-s + 0.353i·8-s + (−0.0910 + 0.995i)9-s + (0.588 + 1.01i)10-s + (0.830 + 0.479i)11-s + (−0.337 − 0.369i)12-s + (0.0954 − 0.165i)13-s + (0.432 − 0.559i)14-s + (−1.58 − 0.503i)15-s + 0.250·16-s + (−1.61 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.488 - 0.872i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.488 - 0.872i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.13973 + 0.667744i\)
\(L(\frac12)\) \(\approx\) \(1.13973 + 0.667744i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (-2.02 - 2.21i)T \)
7 \( 1 + (-5.54 - 4.27i)T \)
good5 \( 1 + (7.20 - 4.15i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-9.13 - 5.27i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.24 + 2.14i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (27.5 - 15.8i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.85 + 11.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-27.8 + 16.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-2.05 + 1.18i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 20.7T + 961T^{2} \)
37 \( 1 + (-5.23 + 9.07i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (43.1 + 24.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-16.3 - 28.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 41.5iT - 2.20e3T^{2} \)
53 \( 1 + (-67.8 + 39.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 71.7iT - 3.48e3T^{2} \)
61 \( 1 - 66.1T + 3.72e3T^{2} \)
67 \( 1 + 24.9T + 4.48e3T^{2} \)
71 \( 1 - 23.3iT - 5.04e3T^{2} \)
73 \( 1 + (-19.5 - 33.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + 10.0T + 6.24e3T^{2} \)
83 \( 1 + (-95.0 + 54.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-36.4 - 21.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (37.7 + 65.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.29502737008958097344420318682, −11.91403498404725115688369070951, −11.21017715378081162202080494669, −10.49139396820355525124810705040, −8.952121795586506291681234568283, −8.345085456120411857360176743106, −6.99428417903623935623319645902, −4.71846229186220626787280125810, −3.85095038901968898432394453874, −2.49000128174386217903419136464, 0.938830310816403100808265153958, 3.71622664109638310700991208744, 4.78388687592185816172239468037, 6.76679707462799335195638715425, 7.60518037700712493655207260505, 8.478941427844273287454767413048, 9.158899914035295598447453827220, 11.30854036113885093216191391119, 11.95658848973617648104743207451, 13.22797059766953120170692921614

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