↑

Properties

Label 2-2151-2151.1672-c0-0-4
Degree $2$
Conductor $2151$
Sign $0.173 - 0.984i$
Analytic cond. $1.07348$
Root an. cond. $1.03609$
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  + (−0.766 + 1.32i)2-s + (−0.5 − 0.866i)3-s + (−0.673 − 1.16i)4-s + (0.939 + 1.62i)5-s + 1.53·6-s + 0.532·8-s + (−0.499 + 0.866i)9-s − 2.87·10-s + (0.939 − 1.62i)11-s + (−0.673 + 1.16i)12-s + (0.939 − 1.62i)15-s + (0.266 − 0.460i)16-s + 1.53·17-s + (−0.766 − 1.32i)18-s + (1.26 − 2.19i)20-s + ⋯
L(s)  = 1  + (−0.766 + 1.32i)2-s + (−0.5 − 0.866i)3-s + (−0.673 − 1.16i)4-s + (0.939 + 1.62i)5-s + 1.53·6-s + 0.532·8-s + (−0.499 + 0.866i)9-s − 2.87·10-s + (0.939 − 1.62i)11-s + (−0.673 + 1.16i)12-s + (0.939 − 1.62i)15-s + (0.266 − 0.460i)16-s + 1.53·17-s + (−0.766 − 1.32i)18-s + (1.26 − 2.19i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1672, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8183400869\)
\(L(\frac12)\) \(\approx\) \(0.8183400869\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
239 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.53T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−9.361058485065056553070451387477, −8.294487114477090580495913409061, −7.80102280649269624761815237627, −6.92488872040319195282621482367, −6.42307748717762087975747249719, −5.92426996802249720006710014724, −5.46012997129127203031257822447, −3.47770832589686805945734261048, −2.57364029131024282744214093932, −1.08051390806981075639607526879, 1.12689212121691924572331689639, 1.74885609067521600487305241119, 3.16309681529490038100986469478, 4.22976821137005796990788698326, 4.89191492470362284593141015459, 5.63432947157410551003597522954, 6.64493471473812664232330985278, 8.040649153968843381438029909141, 8.919646361231060151630386221931, 9.316093041526899046682241126014

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