Properties

Label 2-147-147.101-c3-0-38
Degree $2$
Conductor $147$
Sign $0.702 - 0.711i$
Analytic cond. $8.67328$
Root an. cond. $2.94504$
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  + (4.32 + 1.69i)2-s + (5.19 + 0.184i)3-s + (9.94 + 9.22i)4-s + (−7.56 − 5.15i)5-s + (22.1 + 9.60i)6-s + (16.1 + 9.04i)7-s + (11.2 + 23.3i)8-s + (26.9 + 1.91i)9-s + (−23.9 − 35.1i)10-s + (−4.18 − 27.7i)11-s + (49.9 + 49.7i)12-s + (−55.3 + 44.1i)13-s + (54.5 + 66.5i)14-s + (−38.3 − 28.1i)15-s + (0.864 + 11.5i)16-s + (76.7 + 23.6i)17-s + ⋯
L(s)  = 1  + (1.52 + 0.599i)2-s + (0.999 + 0.0355i)3-s + (1.24 + 1.15i)4-s + (−0.676 − 0.461i)5-s + (1.50 + 0.653i)6-s + (0.872 + 0.488i)7-s + (0.495 + 1.02i)8-s + (0.997 + 0.0710i)9-s + (−0.757 − 1.11i)10-s + (−0.114 − 0.761i)11-s + (1.20 + 1.19i)12-s + (−1.17 + 0.940i)13-s + (1.04 + 1.27i)14-s + (−0.659 − 0.484i)15-s + (0.0135 + 0.180i)16-s + (1.09 + 0.337i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(8.67328\)
Root analytic conductor: \(2.94504\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :3/2),\ 0.702 - 0.711i)\)

Particular Values

\(L(2)\) \(\approx\) \(4.38781 + 1.83468i\)
\(L(\frac12)\) \(\approx\) \(4.38781 + 1.83468i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.19 - 0.184i)T \)
7 \( 1 + (-16.1 - 9.04i)T \)
good2 \( 1 + (-4.32 - 1.69i)T + (5.86 + 5.44i)T^{2} \)
5 \( 1 + (7.56 + 5.15i)T + (45.6 + 116. i)T^{2} \)
11 \( 1 + (4.18 + 27.7i)T + (-1.27e3 + 392. i)T^{2} \)
13 \( 1 + (55.3 - 44.1i)T + (488. - 2.14e3i)T^{2} \)
17 \( 1 + (-76.7 - 23.6i)T + (4.05e3 + 2.76e3i)T^{2} \)
19 \( 1 + (73.4 - 42.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (41.5 + 134. i)T + (-1.00e4 + 6.85e3i)T^{2} \)
29 \( 1 + (275. - 62.9i)T + (2.19e4 - 1.05e4i)T^{2} \)
31 \( 1 + (99.1 + 57.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-172. + 159. i)T + (3.78e3 - 5.05e4i)T^{2} \)
41 \( 1 + (-205. + 98.8i)T + (4.29e4 - 5.38e4i)T^{2} \)
43 \( 1 + (-224. - 108. i)T + (4.95e4 + 6.21e4i)T^{2} \)
47 \( 1 + (145. - 369. i)T + (-7.61e4 - 7.06e4i)T^{2} \)
53 \( 1 + (14.1 - 15.2i)T + (-1.11e4 - 1.48e5i)T^{2} \)
59 \( 1 + (237. - 161. i)T + (7.50e4 - 1.91e5i)T^{2} \)
61 \( 1 + (261. + 281. i)T + (-1.69e4 + 2.26e5i)T^{2} \)
67 \( 1 + (124. - 215. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-240. - 54.9i)T + (3.22e5 + 1.55e5i)T^{2} \)
73 \( 1 + (-560. + 219. i)T + (2.85e5 - 2.64e5i)T^{2} \)
79 \( 1 + (-311. - 540. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (220. - 276. i)T + (-1.27e5 - 5.57e5i)T^{2} \)
89 \( 1 + (397. + 59.9i)T + (6.73e5 + 2.07e5i)T^{2} \)
97 \( 1 - 393. iT - 9.12e5T^{2} \)
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

−12.62676191218701449752334284100, −12.38174978210158088444886456894, −11.09435571750665122291177552700, −9.332468684158743558281549012590, −8.116908660704151640508896154699, −7.48479477877986162171191846202, −5.92971505585395486904429865754, −4.62758274804065962721099338223, −3.88289583352248134236948358403, −2.32676844211270361308166446907, 1.98404555683900058789966101639, 3.25188787358299733033808147133, 4.23737272866526066128703322230, 5.31048223923630698656021116126, 7.31097120145360364940220602994, 7.80188922984658053995285485166, 9.677463433616444416884960665168, 10.73414875765000367927274153941, 11.71481552543677661712967421400, 12.64956155927720059968925074557

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