↑

Properties

Label 2-3639-3639.3638-c0-0-20
Degree $2$
Conductor $3639$
Sign $1$
Analytic cond. $1.81609$
Root an. cond. $1.34762$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·2-s + 3-s − 0.618·4-s + 0.618·5-s + 0.618·6-s + 0.618·7-s − 8-s + 9-s + 0.381·10-s − 0.618·12-s + 0.618·13-s + 0.381·14-s + 0.618·15-s + 0.618·17-s + 0.618·18-s + 0.618·19-s − 0.381·20-s + 0.618·21-s − 1.61·23-s − 24-s − 0.618·25-s + 0.381·26-s + 27-s − 0.381·28-s + 0.381·30-s − 1.61·31-s + 0.999·32-s + ⋯
L(s)  = 1  + 0.618·2-s + 3-s − 0.618·4-s + 0.618·5-s + 0.618·6-s + 0.618·7-s − 8-s + 9-s + 0.381·10-s − 0.618·12-s + 0.618·13-s + 0.381·14-s + 0.618·15-s + 0.618·17-s + 0.618·18-s + 0.618·19-s − 0.381·20-s + 0.618·21-s − 1.61·23-s − 24-s − 0.618·25-s + 0.381·26-s + 27-s − 0.381·28-s + 0.381·30-s − 1.61·31-s + 0.999·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3639\)    =    \(3 \cdot 1213\)
Sign: $1$
Analytic conductor: \(1.81609\)
Root analytic conductor: \(1.34762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3639} (3638, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3639,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.502414037\)
\(L(\frac12)\) \(\approx\) \(2.502414037\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
1213 \( 1 - T \)
good2 \( 1 - 0.618T + T^{2} \)
5 \( 1 - 0.618T + T^{2} \)
7 \( 1 - 0.618T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 0.618T + T^{2} \)
17 \( 1 - 0.618T + T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.61T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + 1.61T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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

−8.798693292057204592309204392586, −7.88051159164201977631954244094, −7.60134127304552928414876452004, −6.17727166806926455045265260549, −5.73424947135752167387386442920, −4.79536462357815651046306960089, −4.00324303559848489646029116731, −3.42366348457728268453665936160, −2.35764837275782028346923206537, −1.40560349032887503086576257804, 1.40560349032887503086576257804, 2.35764837275782028346923206537, 3.42366348457728268453665936160, 4.00324303559848489646029116731, 4.79536462357815651046306960089, 5.73424947135752167387386442920, 6.17727166806926455045265260549, 7.60134127304552928414876452004, 7.88051159164201977631954244094, 8.798693292057204592309204392586

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