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Properties

Label 2-126-63.47-c3-0-11
Degree $2$
Conductor $126$
Sign $0.997 - 0.0771i$
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 + (1.98 + 4.80i)3-s + (1.99 − 3.46i)4-s + 12.8·5-s + (8.23 + 6.34i)6-s + (7.27 − 17.0i)7-s − 7.99i·8-s + (−19.1 + 19.0i)9-s + (22.2 − 12.8i)10-s + 20.7i·11-s + (20.6 + 2.74i)12-s + (6.74 − 3.89i)13-s + (−4.43 − 36.7i)14-s + (25.4 + 61.7i)15-s + (−8 − 13.8i)16-s + (6.76 + 11.7i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.381 + 0.924i)3-s + (0.249 − 0.433i)4-s + 1.15·5-s + (0.560 + 0.431i)6-s + (0.392 − 0.919i)7-s − 0.353i·8-s + (−0.709 + 0.704i)9-s + (0.704 − 0.406i)10-s + 0.569i·11-s + (0.495 + 0.0660i)12-s + (0.143 − 0.0830i)13-s + (−0.0846 − 0.702i)14-s + (0.438 + 1.06i)15-s + (−0.125 − 0.216i)16-s + (0.0965 + 0.167i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.997 - 0.0771i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.997 - 0.0771i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.01647 + 0.116511i\)
\(L(\frac12)\) \(\approx\) \(3.01647 + 0.116511i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.98 - 4.80i)T \)
7 \( 1 + (-7.27 + 17.0i)T \)
good5 \( 1 - 12.8T + 125T^{2} \)
11 \( 1 - 20.7iT - 1.33e3T^{2} \)
13 \( 1 + (-6.74 + 3.89i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-6.76 - 11.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-68.1 - 39.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 23.7iT - 1.21e4T^{2} \)
29 \( 1 + (-32.2 - 18.6i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (230. + 132. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (208. - 361. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (230. + 399. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-115. + 199. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-127. - 221. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-165. + 95.7i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (18.1 - 31.4i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (722. - 416. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-109. + 190. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 394. iT - 3.57e5T^{2} \)
73 \( 1 + (837. - 483. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (357. + 618. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-544. + 942. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (29.6 - 51.3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (177. + 102. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−13.27027930729248904976708048682, −11.83318926339050242505581712782, −10.51669216229052753851973387299, −10.12238952083476364121343298203, −9.012888541476179599289837912065, −7.42784354685422756562579799400, −5.81150697905845236793496367813, −4.77194298658503584490387853999, −3.51704571872492252204090307349, −1.86093818895071971925256329635, 1.78306917981520093163503980623, 3.06490740256454250987674837089, 5.34335795310833790704472167410, 6.05973985582929302988943656615, 7.27349528023405646312881446130, 8.554364425994159789750470179677, 9.403249756569738281818256501865, 11.16236572420860550276922071603, 12.16515809187909941351888213173, 13.08920590640335566459218714510

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