↑

Properties

Label 2-968-11.6-c0-0-1
Degree $2$
Conductor $968$
Sign $0.860 + 0.509i$
Analytic cond. $0.483094$
Root an. cond. $0.695050$
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  + (0.809 + 0.587i)3-s + (0.309 − 0.951i)5-s + (−0.831 − 1.14i)7-s + (0.809 − 0.587i)15-s + (1.34 + 0.437i)17-s − 1.41i·21-s − 23-s + (0.309 − 0.951i)27-s + (0.309 + 0.951i)31-s + (−1.34 + 0.437i)35-s + (−0.809 + 0.587i)37-s + 1.41i·43-s + (−0.309 + 0.951i)49-s + (0.831 + 1.14i)51-s + (−0.809 + 0.587i)59-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (0.309 − 0.951i)5-s + (−0.831 − 1.14i)7-s + (0.809 − 0.587i)15-s + (1.34 + 0.437i)17-s − 1.41i·21-s − 23-s + (0.309 − 0.951i)27-s + (0.309 + 0.951i)31-s + (−1.34 + 0.437i)35-s + (−0.809 + 0.587i)37-s + 1.41i·43-s + (−0.309 + 0.951i)49-s + (0.831 + 1.14i)51-s + (−0.809 + 0.587i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.860 + 0.509i$
Analytic conductor: \(0.483094\)
Root analytic conductor: \(0.695050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :0),\ 0.860 + 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.261263795\)
\(L(\frac12)\) \(\approx\) \(1.261263795\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.965152036017739667978922840690, −9.457096047109454597563968442326, −8.556687198070700965881601681702, −7.87836945283735255069500130937, −6.77819817808999900471909403859, −5.80349043373972800565570004637, −4.66488702355191493000068648470, −3.78567196601666207751237357000, −3.06333174787931476590933791131, −1.26294153761128970126630603489, 2.07539870427375014707055299856, 2.77923755223142338495361035367, 3.57700461717680160225671419087, 5.34667550668267733056026178637, 6.09627226770634438129788895606, 6.97048332615423162853920057865, 7.78095946106249810173537447800, 8.580238719314343228831282009150, 9.474596260095773552269457913470, 10.09173447044792034791304654595

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