↑

Properties

Label 2-2144-536.475-c0-0-0
Degree $2$
Conductor $2144$
Sign $-0.0965 - 0.995i$
Analytic cond. $1.06999$
Root an. cond. $1.03440$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0395 + 0.0865i)3-s + (0.648 + 0.748i)9-s + (−1.21 + 1.16i)11-s + (−1.45 + 1.14i)17-s + (−0.462 − 0.0892i)19-s + (0.841 + 0.540i)25-s + (−0.181 + 0.0533i)27-s + (−0.0523 − 0.151i)33-s + (−1.21 − 0.486i)41-s + (0.205 + 1.43i)43-s + (0.928 − 0.371i)49-s + (−0.0416 − 0.171i)51-s + (0.0260 − 0.0365i)57-s + (1.49 − 0.961i)59-s + (0.959 + 0.281i)67-s + ⋯
L(s)  = 1  + (−0.0395 + 0.0865i)3-s + (0.648 + 0.748i)9-s + (−1.21 + 1.16i)11-s + (−1.45 + 1.14i)17-s + (−0.462 − 0.0892i)19-s + (0.841 + 0.540i)25-s + (−0.181 + 0.0533i)27-s + (−0.0523 − 0.151i)33-s + (−1.21 − 0.486i)41-s + (0.205 + 1.43i)43-s + (0.928 − 0.371i)49-s + (−0.0416 − 0.171i)51-s + (0.0260 − 0.0365i)57-s + (1.49 − 0.961i)59-s + (0.959 + 0.281i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0965 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0965 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2144\)    =    \(2^{5} \cdot 67\)
Sign: $-0.0965 - 0.995i$
Analytic conductor: \(1.06999\)
Root analytic conductor: \(1.03440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2144} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2144,\ (\ :0),\ -0.0965 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9028602354\)
\(L(\frac12)\) \(\approx\) \(0.9028602354\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
67 \( 1 + (-0.959 - 0.281i)T \)
good3 \( 1 + (0.0395 - 0.0865i)T + (-0.654 - 0.755i)T^{2} \)
5 \( 1 + (-0.841 - 0.540i)T^{2} \)
7 \( 1 + (-0.928 + 0.371i)T^{2} \)
11 \( 1 + (1.21 - 1.16i)T + (0.0475 - 0.998i)T^{2} \)
13 \( 1 + (0.995 + 0.0950i)T^{2} \)
17 \( 1 + (1.45 - 1.14i)T + (0.235 - 0.971i)T^{2} \)
19 \( 1 + (0.462 + 0.0892i)T + (0.928 + 0.371i)T^{2} \)
23 \( 1 + (-0.981 + 0.189i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.995 - 0.0950i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.21 + 0.486i)T + (0.723 + 0.690i)T^{2} \)
43 \( 1 + (-0.205 - 1.43i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (0.327 - 0.945i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-1.49 + 0.961i)T + (0.415 - 0.909i)T^{2} \)
61 \( 1 + (-0.0475 - 0.998i)T^{2} \)
71 \( 1 + (-0.235 - 0.971i)T^{2} \)
73 \( 1 + (1.13 + 1.08i)T + (0.0475 + 0.998i)T^{2} \)
79 \( 1 + (-0.580 + 0.814i)T^{2} \)
83 \( 1 + (-0.452 - 1.86i)T + (-0.888 + 0.458i)T^{2} \)
89 \( 1 + (0.271 + 0.595i)T + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.546583307409053523057501915988, −8.606397748699896714658475849555, −7.949873583407999977871627779643, −7.12994456000487356172113015634, −6.51919630980145231880959831090, −5.28621738132721090938730416761, −4.71851758695835714151285669628, −3.92761800268761937711267909295, −2.50276138055233801509138005433, −1.81100369473841582830443090520, 0.61153056346910411153823752451, 2.27231515785679921397887522759, 3.14709480587912828369467837872, 4.21688634975627241841549531695, 5.05821557575046060049469409617, 5.94959877712037776482124216981, 6.80856006097306897515355264988, 7.36427756044630974676527017821, 8.579820621024721789038109558260, 8.785683713949257537691547315486

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