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Properties

Label 2-126-9.4-c3-0-4
Degree $2$
Conductor $126$
Sign $0.0990 - 0.995i$
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 + (4.10 + 3.18i)3-s + (−1.99 + 3.46i)4-s + (−4.11 + 7.12i)5-s + (1.41 − 10.2i)6-s + (−3.5 − 6.06i)7-s + 7.99·8-s + (6.67 + 26.1i)9-s + 16.4·10-s + (4.33 + 7.51i)11-s + (−19.2 + 7.83i)12-s + (−39.6 + 68.7i)13-s + (−7 + 12.1i)14-s + (−39.6 + 16.1i)15-s + (−8 − 13.8i)16-s − 106.·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.789 + 0.613i)3-s + (−0.249 + 0.433i)4-s + (−0.368 + 0.637i)5-s + (0.0964 − 0.700i)6-s + (−0.188 − 0.327i)7-s + 0.353·8-s + (0.247 + 0.968i)9-s + 0.520·10-s + (0.118 + 0.205i)11-s + (−0.463 + 0.188i)12-s + (−0.846 + 1.46i)13-s + (−0.133 + 0.231i)14-s + (−0.681 + 0.277i)15-s + (−0.125 − 0.216i)16-s − 1.52·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0990 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0990 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.0990 - 0.995i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.0990 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.942289 + 0.853149i\)
\(L(\frac12)\) \(\approx\) \(0.942289 + 0.853149i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.10 - 3.18i)T \)
7 \( 1 + (3.5 + 6.06i)T \)
good5 \( 1 + (4.11 - 7.12i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-4.33 - 7.51i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (39.6 - 68.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 106.T + 4.91e3T^{2} \)
19 \( 1 - 137.T + 6.85e3T^{2} \)
23 \( 1 + (75.2 - 130. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-18.7 - 32.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-127. + 220. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 258.T + 5.06e4T^{2} \)
41 \( 1 + (-1.23 + 2.14i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-82.0 - 142. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (245. + 424. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 114.T + 1.48e5T^{2} \)
59 \( 1 + (-181. + 313. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (390. + 675. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (2.38 - 4.12i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 446.T + 3.57e5T^{2} \)
73 \( 1 - 522.T + 3.89e5T^{2} \)
79 \( 1 + (-475. - 824. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-292. - 505. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.13e3T + 7.04e5T^{2} \)
97 \( 1 + (348. + 603. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.34400475965522137101925620844, −11.72872713631295521831695648548, −11.11475372091208332831792551175, −9.719633789494947234933725645061, −9.371265965658186241363105277193, −7.86415027550506126971493186358, −6.95024045169650017922858297346, −4.63886545237429010722338604093, −3.55724993594917796794220690286, −2.18958962001169744457138330973, 0.67625337689599954903522451548, 2.78077798120787801259174691166, 4.68144274175332946476038301126, 6.19347285946995029322293531287, 7.45305780448032136263659411037, 8.308615757563038145297559502308, 9.111160342120241870336764447735, 10.24946959280476432435811744139, 11.95246411526502149247256129273, 12.76782862559621728091907197087

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