Properties

Label 2-2009-1.1-c1-0-112
Degree $2$
Conductor $2009$
Sign $1$
Analytic cond. $16.0419$
Root an. cond. $4.00523$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.73·2-s + 2.52·3-s + 5.48·4-s − 1.62·5-s + 6.89·6-s + 9.52·8-s + 3.36·9-s − 4.45·10-s − 1.21·11-s + 13.8·12-s − 3.49·13-s − 4.10·15-s + 15.0·16-s + 3.59·17-s + 9.19·18-s − 0.898·19-s − 8.92·20-s − 3.32·22-s + 8.47·23-s + 24.0·24-s − 2.35·25-s − 9.55·26-s + 0.915·27-s − 7.30·29-s − 11.2·30-s − 10.0·31-s + 22.1·32-s + ⋯
L(s)  = 1  + 1.93·2-s + 1.45·3-s + 2.74·4-s − 0.727·5-s + 2.81·6-s + 3.36·8-s + 1.12·9-s − 1.40·10-s − 0.367·11-s + 3.99·12-s − 0.969·13-s − 1.06·15-s + 3.76·16-s + 0.871·17-s + 2.16·18-s − 0.206·19-s − 1.99·20-s − 0.709·22-s + 1.76·23-s + 4.90·24-s − 0.470·25-s − 1.87·26-s + 0.176·27-s − 1.35·29-s − 2.05·30-s − 1.80·31-s + 3.92·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2009\)    =    \(7^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(16.0419\)
Root analytic conductor: \(4.00523\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2009,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.452160500\)
\(L(\frac12)\) \(\approx\) \(8.452160500\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 - 2.73T + 2T^{2} \)
3 \( 1 - 2.52T + 3T^{2} \)
5 \( 1 + 1.62T + 5T^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 + 3.49T + 13T^{2} \)
17 \( 1 - 3.59T + 17T^{2} \)
19 \( 1 + 0.898T + 19T^{2} \)
23 \( 1 - 8.47T + 23T^{2} \)
29 \( 1 + 7.30T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 - 0.346T + 37T^{2} \)
43 \( 1 + 6.77T + 43T^{2} \)
47 \( 1 - 0.195T + 47T^{2} \)
53 \( 1 - 5.44T + 53T^{2} \)
59 \( 1 - 8.62T + 59T^{2} \)
61 \( 1 - 0.144T + 61T^{2} \)
67 \( 1 + 2.58T + 67T^{2} \)
71 \( 1 - 7.50T + 71T^{2} \)
73 \( 1 + 5.95T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + 5.77T + 89T^{2} \)
97 \( 1 - 3.67T + 97T^{2} \)
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

−9.064666781377765048182280509134, −7.970046398786115320076478256757, −7.41876820772617765749258739815, −6.98726271147623394542212952382, −5.59891529715385714696554627049, −5.00595868413010612809896176702, −3.93885175419994342214601561100, −3.46784143959195869830516684933, −2.70950414519358876546896179641, −1.85179962762195688291025639052, 1.85179962762195688291025639052, 2.70950414519358876546896179641, 3.46784143959195869830516684933, 3.93885175419994342214601561100, 5.00595868413010612809896176702, 5.59891529715385714696554627049, 6.98726271147623394542212952382, 7.41876820772617765749258739815, 7.970046398786115320076478256757, 9.064666781377765048182280509134

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