↑

Properties

Label 2-147-7.5-c6-0-35
Degree $2$
Conductor $147$
Sign $-0.553 - 0.832i$
Analytic cond. $33.8179$
Root an. cond. $5.81532$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (6.58 − 11.4i)2-s + (−13.5 + 7.79i)3-s + (−54.8 − 94.9i)4-s + (68.9 + 39.7i)5-s + 205. i·6-s − 601.·8-s + (121.5 − 210. i)9-s + (908. − 524. i)10-s + (−411. − 712. i)11-s + (1.48e3 + 854. i)12-s − 2.42e3i·13-s − 1.24e3·15-s + (−456. + 791. i)16-s + (−6.75e3 + 3.89e3i)17-s + (−1.60e3 − 2.77e3i)18-s + (−5.78e3 − 3.34e3i)19-s + ⋯
L(s)  = 1  + (0.823 − 1.42i)2-s + (−0.5 + 0.288i)3-s + (−0.856 − 1.48i)4-s + (0.551 + 0.318i)5-s + 0.951i·6-s − 1.17·8-s + (0.166 − 0.288i)9-s + (0.908 − 0.524i)10-s + (−0.308 − 0.534i)11-s + (0.856 + 0.494i)12-s − 1.10i·13-s − 0.367·15-s + (−0.111 + 0.193i)16-s + (−1.37 + 0.793i)17-s + (−0.274 − 0.475i)18-s + (−0.843 − 0.487i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.553 - 0.832i$
Analytic conductor: \(33.8179\)
Root analytic conductor: \(5.81532\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3),\ -0.553 - 0.832i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.057711650\)
\(L(\frac12)\) \(\approx\) \(1.057711650\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.5 - 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (-6.58 + 11.4i)T + (-32 - 55.4i)T^{2} \)
5 \( 1 + (-68.9 - 39.7i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (411. + 712. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 2.42e3iT - 4.82e6T^{2} \)
17 \( 1 + (6.75e3 - 3.89e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (5.78e3 + 3.34e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (9.41e3 - 1.63e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 1.38e4T + 5.94e8T^{2} \)
31 \( 1 + (-2.41e4 + 1.39e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (3.98e4 - 6.90e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 5.91e4iT - 4.75e9T^{2} \)
43 \( 1 + 9.18e4T + 6.32e9T^{2} \)
47 \( 1 + (-4.34e3 - 2.50e3i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (9.31e4 + 1.61e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (1.95e5 - 1.12e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-1.25e5 - 7.21e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-1.17e5 - 2.03e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 9.62e4T + 1.28e11T^{2} \)
73 \( 1 + (2.38e5 - 1.37e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-3.40e5 + 5.90e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 1.28e5iT - 3.26e11T^{2} \)
89 \( 1 + (-3.22e5 - 1.86e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 6.20e5iT - 8.32e11T^{2} \)
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

−11.24885893632301955901508362505, −10.48408552611911636503995037506, −9.897310258242121138047687161585, −8.369542543947710582810251292599, −6.40306323857905204995808019478, −5.41462800723018202874680117454, −4.26480057395925574136299287944, −3.03994015497324337692851002495, −1.84334088317644376309649117863, −0.23560527128413133065899137184, 2.02367135822062937674664547972, 4.31128505712049518903841859410, 5.01288057857263130695141484498, 6.32801528067315982392323561313, 6.79294899344900546867079684012, 8.086375185121013484294050608580, 9.192148683456870278767680626874, 10.61754649700136008877542849394, 12.02978634605688270016492369218, 12.88149085437214971411992690066

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