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Properties

Label 2-126-9.5-c2-0-6
Degree $2$
Conductor $126$
Sign $0.762 + 0.647i$
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 + (0.530 − 2.95i)3-s + (0.999 − 1.73i)4-s + (5.20 + 3.00i)5-s + (1.43 + 3.99i)6-s + (−1.32 − 2.29i)7-s + 2.82i·8-s + (−8.43 − 3.13i)9-s − 8.49·10-s + (15.5 − 8.98i)11-s + (−4.58 − 3.87i)12-s + (−1.24 + 2.14i)13-s + (3.24 + 1.87i)14-s + (11.6 − 13.7i)15-s + (−2.00 − 3.46i)16-s − 26.3i·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.176 − 0.984i)3-s + (0.249 − 0.433i)4-s + (1.04 + 0.601i)5-s + (0.239 + 0.665i)6-s + (−0.188 − 0.327i)7-s + 0.353i·8-s + (−0.937 − 0.347i)9-s − 0.849·10-s + (1.41 − 0.817i)11-s + (−0.382 − 0.322i)12-s + (−0.0953 + 0.165i)13-s + (0.231 + 0.133i)14-s + (0.775 − 0.918i)15-s + (−0.125 − 0.216i)16-s − 1.55i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.762 + 0.647i$
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: \((2,\ 126,\ (\ :1),\ 0.762 + 0.647i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.21237 - 0.445511i\)
\(L(\frac12)\) \(\approx\) \(1.21237 - 0.445511i\)
\(L(2)\) 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 \( 1 + (-0.530 + 2.95i)T \)
7 \( 1 + (1.32 + 2.29i)T \)
good5 \( 1 + (-5.20 - 3.00i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-15.5 + 8.98i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (1.24 - 2.14i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 26.3iT - 289T^{2} \)
19 \( 1 - 5.05T + 361T^{2} \)
23 \( 1 + (-29.8 - 17.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (17.4 - 10.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-0.457 + 0.792i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 69.8T + 1.36e3T^{2} \)
41 \( 1 + (-43.6 - 25.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-1.22 - 2.11i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (45.5 - 26.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 91.2iT - 2.80e3T^{2} \)
59 \( 1 + (-13.6 - 7.88i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-39.3 - 68.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (1.52 - 2.63i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 58.7iT - 5.04e3T^{2} \)
73 \( 1 + 8.20T + 5.32e3T^{2} \)
79 \( 1 + (23.0 + 39.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-33.1 + 19.1i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 88.0iT - 7.92e3T^{2} \)
97 \( 1 + (-2.91 - 5.04i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.37193334339529886765496076318, −11.84020322651392051907305128639, −10.99204758772384782443521123519, −9.525941302058363140628910566316, −8.927493373956245871595691208439, −7.30522612701717163855199071158, −6.68095764642553061844906985009, −5.63676888467891614541656800840, −2.98393093820099867394785410705, −1.26676934970104611631330563983, 1.88620188879849209049795619234, 3.75563052998505760758670736898, 5.23108259138124709210205166195, 6.58155228903267511197816903595, 8.483099824013365328431731098711, 9.241615079092878482248413552133, 9.881368620293654463003420889139, 10.89337161020430584360992009313, 12.16312561969134875149471627391, 13.08576011230845284726736703406

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