↑

Properties

Label 2-147-21.20-c5-0-49
Degree $2$
Conductor $147$
Sign $-0.842 + 0.538i$
Analytic cond. $23.5764$
Root an. cond. $4.85555$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.81i·2-s + (−6.23 − 14.2i)3-s + 17.4·4-s + 63.5·5-s + (−54.4 + 23.7i)6-s − 188. i·8-s + (−165. + 178. i)9-s − 242. i·10-s − 192. i·11-s + (−108. − 249. i)12-s + 82.7i·13-s + (−396. − 908. i)15-s − 160.·16-s + 1.84e3·17-s + (679. + 630. i)18-s − 2.02e3i·19-s + ⋯
L(s)  = 1  − 0.674i·2-s + (−0.399 − 0.916i)3-s + 0.545·4-s + 1.13·5-s + (−0.617 + 0.269i)6-s − 1.04i·8-s + (−0.680 + 0.732i)9-s − 0.766i·10-s − 0.480i·11-s + (−0.218 − 0.500i)12-s + 0.135i·13-s + (−0.454 − 1.04i)15-s − 0.156·16-s + 1.55·17-s + (0.494 + 0.458i)18-s − 1.28i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.842 + 0.538i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.842 + 0.538i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.354008656\)
\(L(\frac12)\) \(\approx\) \(2.354008656\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (6.23 + 14.2i)T \)
7 \( 1 \)
good2 \( 1 + 3.81iT - 32T^{2} \)
5 \( 1 - 63.5T + 3.12e3T^{2} \)
11 \( 1 + 192. iT - 1.61e5T^{2} \)
13 \( 1 - 82.7iT - 3.71e5T^{2} \)
17 \( 1 - 1.84e3T + 1.41e6T^{2} \)
19 \( 1 + 2.02e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.75e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.54e3iT - 2.05e7T^{2} \)
31 \( 1 - 8.70e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.57e4T + 6.93e7T^{2} \)
41 \( 1 + 1.05e4T + 1.15e8T^{2} \)
43 \( 1 + 6.69e3T + 1.47e8T^{2} \)
47 \( 1 - 1.69e4T + 2.29e8T^{2} \)
53 \( 1 - 2.94e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.26e4T + 7.14e8T^{2} \)
61 \( 1 + 7.69e3iT - 8.44e8T^{2} \)
67 \( 1 - 4.99e4T + 1.35e9T^{2} \)
71 \( 1 + 1.68e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.41e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.57e4T + 3.07e9T^{2} \)
83 \( 1 + 2.44e4T + 3.93e9T^{2} \)
89 \( 1 + 8.01e4T + 5.58e9T^{2} \)
97 \( 1 + 3.67e3iT - 8.58e9T^{2} \)
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

−11.86968643060237013242606098448, −10.76267522778927368979968938412, −10.04355034285062687154212087267, −8.648764193938765500698085642739, −7.18300787200847497811870133455, −6.33535744655639926631935667988, −5.31025705552549542728199260023, −3.03760697325009219582339890690, −1.94445007993280114392858827513, −0.831810515683793668781708527962, 1.75843399542796949002669067401, 3.46196770778296186040611100970, 5.41737450540677950379551676031, 5.71239144411441023176093405284, 7.03946699797080005116538494987, 8.352758222005572765323846692074, 9.801270403686469814714144075560, 10.20382331922900251688614843293, 11.48373249861560136206422434563, 12.39967915146917296852770594776

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