Properties

Label 2-126-7.2-c3-0-3
Degree $2$
Conductor $126$
Sign $0.902 - 0.431i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−7.91 + 13.7i)5-s + (18.3 + 2.59i)7-s − 7.99·8-s + (15.8 + 27.4i)10-s + (25.9 + 44.8i)11-s + 38.8·13-s + (22.8 − 29.1i)14-s + (−8 + 13.8i)16-s + (13.6 + 23.6i)17-s + (−38.2 + 66.2i)19-s + 63.3·20-s + 103.·22-s + (73.6 − 127. i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.708 + 1.22i)5-s + (0.990 + 0.140i)7-s − 0.353·8-s + (0.500 + 0.867i)10-s + (0.710 + 1.23i)11-s + 0.828·13-s + (0.435 − 0.556i)14-s + (−0.125 + 0.216i)16-s + (0.195 + 0.337i)17-s + (−0.461 + 0.800i)19-s + 0.708·20-s + 1.00·22-s + (0.667 − 1.15i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.902 - 0.431i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.902 - 0.431i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.77492 + 0.402713i\)
\(L(\frac12)\) \(\approx\) \(1.77492 + 0.402713i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
7 \( 1 + (-18.3 - 2.59i)T \)
good5 \( 1 + (7.91 - 13.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-25.9 - 44.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 38.8T + 2.19e3T^{2} \)
17 \( 1 + (-13.6 - 23.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (38.2 - 66.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-73.6 + 127. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 240.T + 2.43e4T^{2} \)
31 \( 1 + (-148. - 256. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-80.7 + 139. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 102.T + 6.89e4T^{2} \)
43 \( 1 + 328.T + 7.95e4T^{2} \)
47 \( 1 + (-33.9 + 58.8i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (33.2 + 57.5i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (230. + 400. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (92.6 - 160. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (272. + 472. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 130.T + 3.57e5T^{2} \)
73 \( 1 + (90.6 + 157. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-204. + 354. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 347.T + 5.71e5T^{2} \)
89 \( 1 + (-578. + 1.00e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.61e3T + 9.12e5T^{2} \)
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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

−12.73900063268825839450673290686, −11.82325195824460425619470323739, −10.99945899264658044861204932547, −10.27791018117584180658422368628, −8.749474606451993875632562555088, −7.49449893909172609093459523426, −6.34382316348076456163316352552, −4.61796203369642929044808974706, −3.49780124844820060528400310886, −1.82365742744416616259945670866, 0.958581059815594624918747057432, 3.75092871780324631700017926096, 4.80728460731211950432347962366, 5.95061352504484639123342760477, 7.57156404509733251522889581131, 8.458443446236150373229732401497, 9.157778419961624937348124009170, 11.29750802784685900191233071752, 11.64646361581292279063694305513, 13.11910402495377210444461780066

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