↑

Properties

Label 2-48e2-72.43-c0-0-3
Degree $2$
Conductor $2304$
Sign $0.906 + 0.422i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + 3-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s + 9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + (0.866 − 0.5i)15-s + (−0.866 − 0.5i)21-s + (0.866 − 0.5i)23-s + 27-s + (−0.866 − 0.5i)29-s + (−0.866 + 0.5i)31-s + (−0.5 + 0.866i)33-s − 0.999·35-s + (0.866 − 0.5i)39-s + ⋯
L(s)  = 1  + 3-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s + 9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)13-s + (0.866 − 0.5i)15-s + (−0.866 − 0.5i)21-s + (0.866 − 0.5i)23-s + 27-s + (−0.866 − 0.5i)29-s + (−0.866 + 0.5i)31-s + (−0.5 + 0.866i)33-s − 0.999·35-s + (0.866 − 0.5i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.906 + 0.422i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1663, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.906 + 0.422i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.830849532\)
\(L(\frac12)\) \(\approx\) \(1.830849532\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
good5 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.284446026125059472531219603797, −8.492770073135583255669099636358, −7.59581055971434859629289286710, −6.97129979093545917792429425937, −6.04824374810446674370881438100, −5.15696840438912223360724588341, −4.16578292739703942738486668735, −3.31585607274060785964024531932, −2.37910290514335720822202058616, −1.32825054439069175272083794113, 1.61601654158749595441237871512, 2.67794752637382999723099780366, 3.23800607359521436650949810093, 4.16636189675723222547085089842, 5.60708871737076146614406785605, 6.05745763999122792673598904235, 6.97709562095182620856948453311, 7.69401811679126053834040061233, 8.817461957785495998476960524397, 9.092419518852106723460288262903

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