Properties

Label 2-3640-1.1-c1-0-53
Degree $2$
Conductor $3640$
Sign $-1$
Analytic cond. $29.0655$
Root an. cond. $5.39124$
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  − 0.700·3-s − 5-s + 7-s − 2.50·9-s + 4.66·11-s + 13-s + 0.700·15-s − 5.15·17-s − 1.75·19-s − 0.700·21-s + 4.06·23-s + 25-s + 3.85·27-s − 7.96·29-s − 1.99·31-s − 3.26·33-s − 35-s − 8.63·37-s − 0.700·39-s + 7.22·41-s − 7.71·43-s + 2.50·45-s + 3.96·47-s + 49-s + 3.61·51-s + 10.7·53-s − 4.66·55-s + ⋯
L(s)  = 1  − 0.404·3-s − 0.447·5-s + 0.377·7-s − 0.836·9-s + 1.40·11-s + 0.277·13-s + 0.180·15-s − 1.25·17-s − 0.403·19-s − 0.152·21-s + 0.848·23-s + 0.200·25-s + 0.742·27-s − 1.47·29-s − 0.358·31-s − 0.568·33-s − 0.169·35-s − 1.41·37-s − 0.112·39-s + 1.12·41-s − 1.17·43-s + 0.374·45-s + 0.577·47-s + 0.142·49-s + 0.505·51-s + 1.47·53-s − 0.629·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(29.0655\)
Root analytic conductor: \(5.39124\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3640,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 + 0.700T + 3T^{2} \)
11 \( 1 - 4.66T + 11T^{2} \)
17 \( 1 + 5.15T + 17T^{2} \)
19 \( 1 + 1.75T + 19T^{2} \)
23 \( 1 - 4.06T + 23T^{2} \)
29 \( 1 + 7.96T + 29T^{2} \)
31 \( 1 + 1.99T + 31T^{2} \)
37 \( 1 + 8.63T + 37T^{2} \)
41 \( 1 - 7.22T + 41T^{2} \)
43 \( 1 + 7.71T + 43T^{2} \)
47 \( 1 - 3.96T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 - 6.06T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 7.22T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 2.24T + 73T^{2} \)
79 \( 1 + 5.82T + 79T^{2} \)
83 \( 1 + 5.40T + 83T^{2} \)
89 \( 1 + 8.98T + 89T^{2} \)
97 \( 1 + 5.97T + 97T^{2} \)
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

−8.448183344410665404530740630882, −7.19483944615982863004627126913, −6.80405467068409336760536802224, −5.89862804977074013086616778843, −5.21323389393103499538876541424, −4.22321341266488313414382565560, −3.66709176799971323918332489009, −2.47618769310528413117324280590, −1.35637210181047361186257302461, 0, 1.35637210181047361186257302461, 2.47618769310528413117324280590, 3.66709176799971323918332489009, 4.22321341266488313414382565560, 5.21323389393103499538876541424, 5.89862804977074013086616778843, 6.80405467068409336760536802224, 7.19483944615982863004627126913, 8.448183344410665404530740630882

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