↑

Properties

Label 2-147-7.4-c5-0-29
Degree $2$
Conductor $147$
Sign $0.386 - 0.922i$
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  + (−2.5 − 4.33i)2-s + (−4.5 + 7.79i)3-s + (3.49 − 6.06i)4-s + (−47 − 81.4i)5-s + 45.0·6-s − 194.·8-s + (−40.5 − 70.1i)9-s + (−235 + 407. i)10-s + (−26 + 45.0i)11-s + (31.4 + 54.5i)12-s − 770·13-s + 846·15-s + (375.5 + 650. i)16-s + (1.01e3 − 1.75e3i)17-s + (−202. + 350. i)18-s + (−866 − 1.49e3i)19-s + ⋯
L(s)  = 1  + (−0.441 − 0.765i)2-s + (−0.288 + 0.499i)3-s + (0.109 − 0.189i)4-s + (−0.840 − 1.45i)5-s + 0.510·6-s − 1.07·8-s + (−0.166 − 0.288i)9-s + (−0.743 + 1.28i)10-s + (−0.0647 + 0.112i)11-s + (0.0631 + 0.109i)12-s − 1.26·13-s + 0.970·15-s + (0.366 + 0.635i)16-s + (0.848 − 1.46i)17-s + (−0.147 + 0.255i)18-s + (−0.550 − 0.953i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(23.5764\)
Root analytic conductor: \(4.85555\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :5/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.09321076783\)
\(L(\frac12)\) \(\approx\) \(0.09321076783\)
\(L(\frac{7}{2})\) 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 \( 1 + (2.5 + 4.33i)T + (-16 + 27.7i)T^{2} \)
5 \( 1 + (47 + 81.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (26 - 45.0i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 770T + 3.71e5T^{2} \)
17 \( 1 + (-1.01e3 + 1.75e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (866 + 1.49e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-288 - 498. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 5.51e3T + 2.05e7T^{2} \)
31 \( 1 + (3.16e3 - 5.48e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-3.66e3 - 6.35e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 3.26e3T + 1.15e8T^{2} \)
43 \( 1 - 5.42e3T + 1.47e8T^{2} \)
47 \( 1 + (432 + 748. i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (2.09e3 - 3.62e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-5.61e3 + 9.71e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.28e4 - 3.94e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (698 - 1.20e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 1.87e4T + 1.80e9T^{2} \)
73 \( 1 + (2.31e4 - 4.01e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.87e4 + 8.43e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 8.12e4T + 3.93e9T^{2} \)
89 \( 1 + (-1.59e3 - 2.75e3i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 4.91e3T + 8.58e9T^{2} \)
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

−11.52460991830108902591220626859, −10.20775060037073260705031894150, −9.393734173172777408380313104588, −8.572608919347088302204634300898, −7.15184180277727195821398170704, −5.32013618168914726815838784354, −4.58876946895826336197408210319, −2.86673435885173735729839925040, −1.00042827893304336138206861675, −0.04592000147254052153412775290, 2.49695535570164848499137492876, 3.76945438269028129350314736482, 5.91285171715296825542745164954, 6.78560937301358583599943218237, 7.65236433931643265227061778090, 8.220666717053406790175624404049, 9.986307054439698984926423031823, 10.96468068133728923207749633309, 12.01658386606749420982568866269, 12.63903704006817609719803203549

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