Properties

Label 2-126-7.4-c3-0-3
Degree $2$
Conductor $126$
Sign $-0.746 - 0.665i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (10.4 + 18.0i)5-s + (−18.3 − 2.59i)7-s − 7.99·8-s + (−20.8 + 36.0i)10-s + (7.58 − 13.1i)11-s + 2.16·13-s + (−13.8 − 34.3i)14-s + (−8 − 13.8i)16-s + (−59.6 + 103. i)17-s + (16.7 + 29.0i)19-s − 83.3·20-s + 30.3·22-s + (0.325 + 0.564i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.931 + 1.61i)5-s + (−0.990 − 0.140i)7-s − 0.353·8-s + (−0.658 + 1.14i)10-s + (0.207 − 0.359i)11-s + 0.0461·13-s + (−0.264 − 0.655i)14-s + (−0.125 − 0.216i)16-s + (−0.851 + 1.47i)17-s + (0.202 + 0.350i)19-s − 0.931·20-s + 0.293·22-s + (0.00295 + 0.00511i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.746 - 0.665i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ -0.746 - 0.665i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.631777 + 1.65900i\)
\(L(\frac12)\) \(\approx\) \(0.631777 + 1.65900i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.4 - 18.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-7.58 + 13.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 2.16T + 2.19e3T^{2} \)
17 \( 1 + (59.6 - 103. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-16.7 - 29.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-0.325 - 0.564i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 163.T + 2.43e4T^{2} \)
31 \( 1 + (-111. + 193. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (84.2 + 145. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 323.T + 6.89e4T^{2} \)
43 \( 1 - 221.T + 7.95e4T^{2} \)
47 \( 1 + (-254. - 439. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (88.2 - 152. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-227. + 393. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (19.3 + 33.4i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (70.8 - 122. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 602.T + 3.57e5T^{2} \)
73 \( 1 + (-551. + 954. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-58.1 - 100. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 568.T + 5.71e5T^{2} \)
89 \( 1 + (191. + 331. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 334.T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.56092324880865144168510709421, −12.59369768941885836068772365689, −11.02757269635093439360533229364, −10.21908501201864766296337681290, −9.156192884005659257655651105353, −7.56260190898312750518822716778, −6.30217464708227521099741933644, −6.10982176661184458065006393916, −3.85554518474414186548152917028, −2.59431671091595671232273793524, 0.837399872145892182213605439823, 2.54644393266645707819597182219, 4.47008788633277261006800360916, 5.41714158738784248193475557107, 6.70577882679436627329826763631, 8.758267584146240670953131384352, 9.370629430800718094420484922995, 10.22179789732593242229622771578, 11.85621648879958059321310281181, 12.54528114968269669527059612784

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