↑

Properties

Label 2-2576-161.48-c0-0-0
Degree $2$
Conductor $2576$
Sign $0.862 + 0.506i$
Analytic cond. $1.28559$
Root an. cond. $1.13383$
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  + (0.654 − 0.755i)7-s + (−0.959 + 0.281i)9-s + (0.239 − 0.153i)11-s + (0.959 + 0.281i)23-s + (0.841 + 0.540i)25-s + (0.698 − 1.53i)29-s + (1.25 − 0.368i)37-s + (−0.273 − 1.89i)43-s + (−0.142 − 0.989i)49-s + (−1.10 + 1.27i)53-s + (−0.415 + 0.909i)63-s + (−0.698 − 0.449i)67-s + (0.239 + 0.153i)71-s + (0.0405 − 0.281i)77-s + (1.10 + 1.27i)79-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)7-s + (−0.959 + 0.281i)9-s + (0.239 − 0.153i)11-s + (0.959 + 0.281i)23-s + (0.841 + 0.540i)25-s + (0.698 − 1.53i)29-s + (1.25 − 0.368i)37-s + (−0.273 − 1.89i)43-s + (−0.142 − 0.989i)49-s + (−1.10 + 1.27i)53-s + (−0.415 + 0.909i)63-s + (−0.698 − 0.449i)67-s + (0.239 + 0.153i)71-s + (0.0405 − 0.281i)77-s + (1.10 + 1.27i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $0.862 + 0.506i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ 0.862 + 0.506i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.251555320\)
\(L(\frac12)\) \(\approx\) \(1.251555320\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
good3 \( 1 + (0.959 - 0.281i)T^{2} \)
5 \( 1 + (-0.841 - 0.540i)T^{2} \)
11 \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
31 \( 1 + (0.959 + 0.281i)T^{2} \)
37 \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.273 + 1.89i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \)
59 \( 1 + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (0.959 + 0.281i)T^{2} \)
67 \( 1 + (0.698 + 0.449i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.239 - 0.153i)T + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.959 - 0.281i)T^{2} \)
97 \( 1 + (-0.841 - 0.540i)T^{2} \)
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

−8.936164468507284085106944261538, −8.210935964156892185003404720681, −7.57931014183801098138520031533, −6.76823755292522251807927365088, −5.87031118072455753810867236447, −5.06476033387860507175572275229, −4.29418596908107989941081453493, −3.29772351670634653311982787898, −2.32029123447014045477016489030, −0.972788070586442169679869966234, 1.29109435694375307851471245811, 2.61250824555255549824477473052, 3.23015965170973645336715208903, 4.64779563263651610540928335591, 5.09964473154496891267550592631, 6.13423265169856030265048764177, 6.67585711259028401431391325939, 7.79046451540367529480965718641, 8.484541517562006646989334735488, 8.982499839803303541410482937256

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