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Properties

Label 2-126-63.32-c2-0-11
Degree $2$
Conductor $126$
Sign $0.714 + 0.699i$
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 + (−2.97 + 0.371i)3-s + (0.999 + 1.73i)4-s − 8.56i·5-s + (−3.90 − 1.64i)6-s + (4.01 − 5.73i)7-s + 2.82i·8-s + (8.72 − 2.21i)9-s + (6.05 − 10.4i)10-s + 7.05i·11-s + (−3.62 − 4.78i)12-s + (7.43 − 12.8i)13-s + (8.97 − 4.18i)14-s + (3.18 + 25.4i)15-s + (−2.00 + 3.46i)16-s + (−12.4 − 7.18i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.992 + 0.123i)3-s + (0.249 + 0.433i)4-s − 1.71i·5-s + (−0.651 − 0.274i)6-s + (0.573 − 0.819i)7-s + 0.353i·8-s + (0.969 − 0.245i)9-s + (0.605 − 1.04i)10-s + 0.641i·11-s + (−0.301 − 0.398i)12-s + (0.571 − 0.990i)13-s + (0.640 − 0.298i)14-s + (0.212 + 1.69i)15-s + (−0.125 + 0.216i)16-s + (−0.732 − 0.422i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.714 + 0.699i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ 0.714 + 0.699i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.36888 - 0.558493i\)
\(L(\frac12)\) \(\approx\) \(1.36888 - 0.558493i\)
\(L(2)\) 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 \( 1 + (2.97 - 0.371i)T \)
7 \( 1 + (-4.01 + 5.73i)T \)
good5 \( 1 + 8.56iT - 25T^{2} \)
11 \( 1 - 7.05iT - 121T^{2} \)
13 \( 1 + (-7.43 + 12.8i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (12.4 + 7.18i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.66 - 16.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + 39.3iT - 529T^{2} \)
29 \( 1 + (11.7 - 6.80i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-12.0 - 20.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-17.6 - 30.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-7.79 - 4.50i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-32.4 - 56.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-28.9 - 16.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-52.4 - 30.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (62.6 - 36.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-3.99 + 6.91i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-41.8 - 72.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 61.0iT - 5.04e3T^{2} \)
73 \( 1 + (9.83 - 17.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-5.09 + 8.82i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-15.8 + 9.15i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (40.0 - 23.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (49.1 + 85.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.79931684698348193201809758737, −12.32446245054543953597406368564, −11.15439312928044761680322391456, −10.03775671315888042179308187997, −8.562894345754972877089413259801, −7.47998545251227795980294746690, −6.00966672174142014502513408449, −4.86991588381691882154501112322, −4.30236515473478709739661559302, −1.05861427945241083459247731185, 2.18301542618570927895562844812, 3.86367861381910705725530066262, 5.54634462409805606387295749554, 6.38677222243108101045271158029, 7.40732850669595698015930423675, 9.346951332945848526693499131445, 10.77153653273469338678598094478, 11.30415002070760890947208681400, 11.79793715259879522570290088743, 13.35842981852306864682135232397

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