↑

Properties

Label 2-147-7.2-c5-0-25
Degree $2$
Conductor $147$
Sign $-0.749 + 0.661i$
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  + (2.65 − 4.60i)2-s + (−4.5 − 7.79i)3-s + (1.85 + 3.21i)4-s + (−51.7 + 89.6i)5-s − 47.8·6-s + 189.·8-s + (−40.5 + 70.1i)9-s + (275. + 476. i)10-s + (−326. − 565. i)11-s + (16.7 − 28.9i)12-s − 138.·13-s + 931.·15-s + (445. − 771. i)16-s + (−587. − 1.01e3i)17-s + (215. + 373. i)18-s + (855. − 1.48e3i)19-s + ⋯
L(s)  = 1  + (0.470 − 0.814i)2-s + (−0.288 − 0.499i)3-s + (0.0580 + 0.100i)4-s + (−0.925 + 1.60i)5-s − 0.542·6-s + 1.04·8-s + (−0.166 + 0.288i)9-s + (0.870 + 1.50i)10-s + (−0.813 − 1.40i)11-s + (0.0335 − 0.0580i)12-s − 0.226·13-s + 1.06·15-s + (0.435 − 0.753i)16-s + (−0.492 − 0.853i)17-s + (0.156 + 0.271i)18-s + (0.543 − 0.941i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ -0.749 + 0.661i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.273751162\)
\(L(\frac12)\) \(\approx\) \(1.273751162\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.5 + 7.79i)T \)
7 \( 1 \)
good2 \( 1 + (-2.65 + 4.60i)T + (-16 - 27.7i)T^{2} \)
5 \( 1 + (51.7 - 89.6i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (326. + 565. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 138.T + 3.71e5T^{2} \)
17 \( 1 + (587. + 1.01e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-855. + 1.48e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-2.01e3 + 3.48e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 2.64e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.43e3 - 2.48e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (1.42e3 - 2.47e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 216.T + 1.15e8T^{2} \)
43 \( 1 - 2.92e3T + 1.47e8T^{2} \)
47 \( 1 + (7.40e3 - 1.28e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.05e4 + 1.83e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.73e4 + 3.00e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-4.37e3 + 7.58e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-6.03e3 - 1.04e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 3.55e4T + 1.80e9T^{2} \)
73 \( 1 + (1.67e4 + 2.89e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.15e4 - 3.73e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 4.33e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.17e4 + 8.96e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 8.62e4T + 8.58e9T^{2} \)
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

−11.42314676261879136792441124739, −11.19312382157626406640770005571, −10.36988035778178417014143761139, −8.326810877135757582783662210331, −7.33620196624619475520988359600, −6.55232876845199477242302151742, −4.74707259453096938848092460935, −3.09726449093255500350114947307, −2.72412054920661003792372208187, −0.39495248597323640326961296751, 1.42179358906136849326341339623, 4.09347879943969210417810120455, 4.86553234405068174818426187293, 5.63589218510323765659797017653, 7.29966270402941915899850452148, 8.079906684961697197383290918562, 9.396538024973260549548700287296, 10.44570127979741034418033789949, 11.74721281056192084649886897238, 12.61888862223956870772958866023

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