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Properties

Label 2-126-63.20-c3-0-5
Degree $2$
Conductor $126$
Sign $0.366 - 0.930i$
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  + (1.73 + i)2-s + (−2.86 − 4.33i)3-s + (1.99 + 3.46i)4-s + (4.49 + 7.78i)5-s + (−0.626 − 10.3i)6-s + (−13.8 + 12.2i)7-s + 7.99i·8-s + (−10.5 + 24.8i)9-s + 17.9i·10-s + (57.4 + 33.1i)11-s + (9.28 − 18.5i)12-s + (42.3 − 24.4i)13-s + (−36.2 + 7.45i)14-s + (20.8 − 41.7i)15-s + (−8 + 13.8i)16-s − 18.7·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.551 − 0.834i)3-s + (0.249 + 0.433i)4-s + (0.401 + 0.696i)5-s + (−0.0426 − 0.705i)6-s + (−0.747 + 0.664i)7-s + 0.353i·8-s + (−0.392 + 0.919i)9-s + 0.568i·10-s + (1.57 + 0.909i)11-s + (0.223 − 0.447i)12-s + (0.904 − 0.521i)13-s + (−0.692 + 0.142i)14-s + (0.359 − 0.719i)15-s + (−0.125 + 0.216i)16-s − 0.267·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.366 - 0.930i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.366 - 0.930i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.56896 + 1.06859i\)
\(L(\frac12)\) \(\approx\) \(1.56896 + 1.06859i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 - i)T \)
3 \( 1 + (2.86 + 4.33i)T \)
7 \( 1 + (13.8 - 12.2i)T \)
good5 \( 1 + (-4.49 - 7.78i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-57.4 - 33.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-42.3 + 24.4i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 18.7T + 4.91e3T^{2} \)
19 \( 1 - 109. iT - 6.85e3T^{2} \)
23 \( 1 + (30.4 - 17.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (53.0 + 30.6i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (289. - 167. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 251.T + 5.06e4T^{2} \)
41 \( 1 + (134. + 232. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-69.7 + 120. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-57.9 + 100. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 438. iT - 1.48e5T^{2} \)
59 \( 1 + (113. + 195. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-469. - 271. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-307. - 531. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 116. iT - 3.57e5T^{2} \)
73 \( 1 - 352. iT - 3.89e5T^{2} \)
79 \( 1 + (-390. + 677. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-122. + 212. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.64e3T + 7.04e5T^{2} \)
97 \( 1 + (-969. - 559. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.94958839289544627777805767295, −12.30838616185212470129011800836, −11.37394291968095828145932003074, −10.10119300418737909534267049392, −8.684904029745831434985506529511, −7.17011220489393632352474252694, −6.38382894016077686015789101776, −5.64601259348265870863174079576, −3.67083704684311702846013978327, −1.94417216215727291714817808943, 0.935498840146053295768408029779, 3.53886216596268599639782916455, 4.40956370008340378310306960145, 5.86849121640263702437333628001, 6.65333785923274232804384540568, 9.069305385659014521709412337915, 9.447620863331514158227065545680, 10.99460515759162266820405999504, 11.42516023126747317550466650292, 12.79269636458651917834122918053

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