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Properties

Label 2-126-63.59-c3-0-15
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: $\chi_{126} (59, \cdot )$
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

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

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