Properties

Label 2-63e2-1.1-c1-0-132
Degree $2$
Conductor $3969$
Sign $-1$
Analytic cond. $31.6926$
Root an. cond. $5.62962$
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  − 0.456·2-s − 1.79·4-s + 4.37·5-s + 1.73·8-s − 1.99·10-s − 2.64·11-s − 4·13-s + 2.79·16-s − 3.46·17-s + 3.58·19-s − 7.84·20-s + 1.20·22-s − 3.46·23-s + 14.1·25-s + 1.82·26-s + 1.82·29-s − 9.16·31-s − 4.73·32-s + 1.58·34-s + 3·37-s − 1.63·38-s + 7.58·40-s − 4.37·41-s − 8.58·43-s + 4.73·44-s + 1.58·46-s − 2.74·47-s + ⋯
L(s)  = 1  − 0.323·2-s − 0.895·4-s + 1.95·5-s + 0.612·8-s − 0.632·10-s − 0.797·11-s − 1.10·13-s + 0.697·16-s − 0.840·17-s + 0.821·19-s − 1.75·20-s + 0.257·22-s − 0.722·23-s + 2.83·25-s + 0.358·26-s + 0.339·29-s − 1.64·31-s − 0.837·32-s + 0.271·34-s + 0.493·37-s − 0.265·38-s + 1.19·40-s − 0.683·41-s − 1.30·43-s + 0.714·44-s + 0.233·46-s − 0.399·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(31.6926\)
Root analytic conductor: \(5.62962\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3969,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good2 \( 1 + 0.456T + 2T^{2} \)
5 \( 1 - 4.37T + 5T^{2} \)
11 \( 1 + 2.64T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 3.58T + 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 1.82T + 29T^{2} \)
31 \( 1 + 9.16T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + 4.37T + 41T^{2} \)
43 \( 1 + 8.58T + 43T^{2} \)
47 \( 1 + 2.74T + 47T^{2} \)
53 \( 1 + 8.66T + 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 + 2.41T + 61T^{2} \)
67 \( 1 + 0.582T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 3.16T + 73T^{2} \)
79 \( 1 + 8.58T + 79T^{2} \)
83 \( 1 + 6.20T + 83T^{2} \)
89 \( 1 + 8.75T + 89T^{2} \)
97 \( 1 + 7.58T + 97T^{2} \)
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

−8.252266099569099241061049267792, −7.39887898271222412069171326259, −6.58876183615029362041872624048, −5.67548358980191560063788235652, −5.16053292904869965214499364826, −4.61291145967534222187845065040, −3.23402317744906237772790240912, −2.27403512392638431120037072497, −1.52236847966008350898464181884, 0, 1.52236847966008350898464181884, 2.27403512392638431120037072497, 3.23402317744906237772790240912, 4.61291145967534222187845065040, 5.16053292904869965214499364826, 5.67548358980191560063788235652, 6.58876183615029362041872624048, 7.39887898271222412069171326259, 8.252266099569099241061049267792

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