Properties

Label 2-147-147.101-c3-0-19
Degree $2$
Conductor $147$
Sign $-0.0195 + 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.47 − 0.971i)2-s + (−4.43 − 2.71i)3-s + (−0.685 − 0.635i)4-s + (1.93 + 1.31i)5-s + (8.33 + 11.0i)6-s + (18.3 − 2.32i)7-s + (10.3 + 21.3i)8-s + (12.2 + 24.0i)9-s + (−3.50 − 5.13i)10-s + (1.51 + 10.0i)11-s + (1.31 + 4.67i)12-s + (20.5 − 16.3i)13-s + (−47.7 − 12.0i)14-s + (−4.98 − 11.0i)15-s + (−4.15 − 55.4i)16-s + (−50.7 − 15.6i)17-s + ⋯
L(s)  = 1  + (−0.874 − 0.343i)2-s + (−0.852 − 0.522i)3-s + (−0.0856 − 0.0794i)4-s + (0.172 + 0.117i)5-s + (0.566 + 0.749i)6-s + (0.992 − 0.125i)7-s + (0.455 + 0.945i)8-s + (0.454 + 0.890i)9-s + (−0.110 − 0.162i)10-s + (0.0414 + 0.275i)11-s + (0.0315 + 0.112i)12-s + (0.438 − 0.349i)13-s + (−0.911 − 0.230i)14-s + (−0.0858 − 0.190i)15-s + (−0.0649 − 0.867i)16-s + (−0.724 − 0.223i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0195 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0195 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.0195 + 0.999i$
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: \((2,\ 147,\ (\ :3/2),\ -0.0195 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.545916 - 0.556696i\)
\(L(\frac12)\) \(\approx\) \(0.545916 - 0.556696i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.43 + 2.71i)T \)
7 \( 1 + (-18.3 + 2.32i)T \)
good2 \( 1 + (2.47 + 0.971i)T + (5.86 + 5.44i)T^{2} \)
5 \( 1 + (-1.93 - 1.31i)T + (45.6 + 116. i)T^{2} \)
11 \( 1 + (-1.51 - 10.0i)T + (-1.27e3 + 392. i)T^{2} \)
13 \( 1 + (-20.5 + 16.3i)T + (488. - 2.14e3i)T^{2} \)
17 \( 1 + (50.7 + 15.6i)T + (4.05e3 + 2.76e3i)T^{2} \)
19 \( 1 + (-92.1 + 53.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-26.5 - 86.1i)T + (-1.00e4 + 6.85e3i)T^{2} \)
29 \( 1 + (-12.1 + 2.76i)T + (2.19e4 - 1.05e4i)T^{2} \)
31 \( 1 + (36.5 + 21.1i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-50.7 + 47.1i)T + (3.78e3 - 5.05e4i)T^{2} \)
41 \( 1 + (-285. + 137. i)T + (4.29e4 - 5.38e4i)T^{2} \)
43 \( 1 + (218. + 105. i)T + (4.95e4 + 6.21e4i)T^{2} \)
47 \( 1 + (9.58 - 24.4i)T + (-7.61e4 - 7.06e4i)T^{2} \)
53 \( 1 + (-319. + 344. i)T + (-1.11e4 - 1.48e5i)T^{2} \)
59 \( 1 + (502. - 342. i)T + (7.50e4 - 1.91e5i)T^{2} \)
61 \( 1 + (-225. - 242. i)T + (-1.69e4 + 2.26e5i)T^{2} \)
67 \( 1 + (-285. + 494. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (360. + 82.2i)T + (3.22e5 + 1.55e5i)T^{2} \)
73 \( 1 + (-932. + 365. i)T + (2.85e5 - 2.64e5i)T^{2} \)
79 \( 1 + (-134. - 232. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-896. + 1.12e3i)T + (-1.27e5 - 5.57e5i)T^{2} \)
89 \( 1 + (108. + 16.3i)T + (6.73e5 + 2.07e5i)T^{2} \)
97 \( 1 - 352. iT - 9.12e5T^{2} \)
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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

−11.91252664722754886103031354490, −11.18807485939647163459071128513, −10.47614699338504459272140364915, −9.332830532519052577915782287062, −8.125855034648816164376836796699, −7.20830894169386052250701584283, −5.66169809704781230958695580252, −4.68992428619370368964098841429, −2.01671356334452871143778622720, −0.73127929741267582483127959616, 1.15435707937296412966068961456, 3.94097597628362267459930907218, 5.12432500964317484251877984430, 6.44191028533274741971576071777, 7.71378611419452270782128155536, 8.799946629397199924126704834381, 9.635560729097719892963306677161, 10.75845882816665663151053812624, 11.54608143504634482983110654074, 12.68800907560548384814344890102

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