Properties

Label 2-3381-1.1-c1-0-123
Degree $2$
Conductor $3381$
Sign $-1$
Analytic cond. $26.9974$
Root an. cond. $5.19590$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.795·2-s + 3-s − 1.36·4-s + 1.31·5-s − 0.795·6-s + 2.67·8-s + 9-s − 1.04·10-s + 4.89·11-s − 1.36·12-s − 5.51·13-s + 1.31·15-s + 0.604·16-s − 3.91·17-s − 0.795·18-s − 4.67·19-s − 1.80·20-s − 3.89·22-s + 23-s + 2.67·24-s − 3.26·25-s + 4.38·26-s + 27-s + 6.49·29-s − 1.04·30-s − 6.46·31-s − 5.83·32-s + ⋯
L(s)  = 1  − 0.562·2-s + 0.577·3-s − 0.683·4-s + 0.589·5-s − 0.324·6-s + 0.946·8-s + 0.333·9-s − 0.331·10-s + 1.47·11-s − 0.394·12-s − 1.52·13-s + 0.340·15-s + 0.151·16-s − 0.950·17-s − 0.187·18-s − 1.07·19-s − 0.403·20-s − 0.830·22-s + 0.208·23-s + 0.546·24-s − 0.652·25-s + 0.860·26-s + 0.192·27-s + 1.20·29-s − 0.191·30-s − 1.16·31-s − 1.03·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(26.9974\)
Root analytic conductor: \(5.19590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3381,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
23 \( 1 - T \)
good2 \( 1 + 0.795T + 2T^{2} \)
5 \( 1 - 1.31T + 5T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 + 5.51T + 13T^{2} \)
17 \( 1 + 3.91T + 17T^{2} \)
19 \( 1 + 4.67T + 19T^{2} \)
29 \( 1 - 6.49T + 29T^{2} \)
31 \( 1 + 6.46T + 31T^{2} \)
37 \( 1 + 9.60T + 37T^{2} \)
41 \( 1 + 8.54T + 41T^{2} \)
43 \( 1 - 3.92T + 43T^{2} \)
47 \( 1 - 8.34T + 47T^{2} \)
53 \( 1 + 5.18T + 53T^{2} \)
59 \( 1 - 1.03T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 3.62T + 71T^{2} \)
73 \( 1 + 16.2T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 3.53T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 - 4.67T + 97T^{2} \)
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

−8.548027498944280223733285716290, −7.55589138118451071346438369593, −6.93142671724064819489580725034, −6.13853330855679712251027728912, −4.96434836338246721922490337620, −4.40382974950925940596096544231, −3.55770960855749017511813274755, −2.26121647272640605631100316673, −1.54876122945227292144329065789, 0, 1.54876122945227292144329065789, 2.26121647272640605631100316673, 3.55770960855749017511813274755, 4.40382974950925940596096544231, 4.96434836338246721922490337620, 6.13853330855679712251027728912, 6.93142671724064819489580725034, 7.55589138118451071346438369593, 8.548027498944280223733285716290

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