↑

Properties

Label 2-147-147.101-c1-0-9
Degree $2$
Conductor $147$
Sign $0.920 + 0.390i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.72 + 0.129i)3-s + (−1.46 − 1.36i)4-s + (2.34 − 1.22i)7-s + (2.96 + 0.447i)9-s + (−2.35 − 2.53i)12-s + (−1.15 + 0.920i)13-s + (0.298 + 3.98i)16-s + (−7.54 + 4.35i)19-s + (4.20 − 1.81i)21-s + (−1.82 − 4.65i)25-s + (5.06 + 1.15i)27-s + (−5.10 − 1.38i)28-s + (4.24 + 2.44i)31-s + (−3.74 − 4.69i)36-s + (−8.50 + 7.88i)37-s + ⋯
L(s)  = 1  + (0.997 + 0.0747i)3-s + (−0.733 − 0.680i)4-s + (0.885 − 0.464i)7-s + (0.988 + 0.149i)9-s + (−0.680 − 0.733i)12-s + (−0.320 + 0.255i)13-s + (0.0747 + 0.997i)16-s + (−1.73 + 0.999i)19-s + (0.917 − 0.397i)21-s + (−0.365 − 0.930i)25-s + (0.974 + 0.222i)27-s + (−0.965 − 0.261i)28-s + (0.761 + 0.439i)31-s + (−0.623 − 0.781i)36-s + (−1.39 + 1.29i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.920 + 0.390i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.920 + 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32575 - 0.269864i\)
\(L(\frac12)\) \(\approx\) \(1.32575 - 0.269864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.129i)T \)
7 \( 1 + (-2.34 + 1.22i)T \)
good2 \( 1 + (1.46 + 1.36i)T^{2} \)
5 \( 1 + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (1.15 - 0.920i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (7.54 - 4.35i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-4.24 - 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.50 - 7.88i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (5.73 + 2.76i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-10.3 - 11.1i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (-7.58 + 13.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-9.58 + 3.76i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (7.35 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (85.0 + 26.2i)T^{2} \)
97 \( 1 + 2.30iT - 97T^{2} \)
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

−13.34217286252171949511818611472, −12.14854341061072267378737592492, −10.54131664177426735776946275383, −10.02329424793354595847315371457, −8.667309837106826278751645616283, −8.125889847795350262269911395521, −6.59684095739916195653670968791, −4.88761665065447289985378879585, −3.97497689942225308427758058086, −1.83137161703910138778855893480, 2.37685516376268386841587932297, 3.92158105218056080153183780612, 5.04702959729112237085809109271, 7.05264662818614947873947440378, 8.196577024922316429304298516019, 8.718691775777919934409993851592, 9.751026715796099584702035756323, 11.16901111313946778139482096138, 12.44348405425755067583770739715, 13.12820913438147036932308618408

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