Properties

Label 2-3640-1.1-c1-0-2
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.43·3-s − 5-s + 7-s − 0.950·9-s − 13-s + 1.43·15-s − 3.95·17-s − 5.95·19-s − 1.43·21-s − 5.03·23-s + 25-s + 5.65·27-s + 4.46·29-s + 2.56·31-s − 35-s + 5.60·37-s + 1.43·39-s − 3.95·41-s − 1.13·43-s + 0.950·45-s + 9.03·47-s + 49-s + 5.65·51-s + 2·53-s + 8.51·57-s − 0.294·59-s − 10·61-s + ⋯
L(s)  = 1  − 0.826·3-s − 0.447·5-s + 0.377·7-s − 0.316·9-s − 0.277·13-s + 0.369·15-s − 0.958·17-s − 1.36·19-s − 0.312·21-s − 1.05·23-s + 0.200·25-s + 1.08·27-s + 0.829·29-s + 0.461·31-s − 0.169·35-s + 0.921·37-s + 0.229·39-s − 0.616·41-s − 0.173·43-s + 0.141·45-s + 1.31·47-s + 0.142·49-s + 0.791·51-s + 0.274·53-s + 1.12·57-s − 0.0383·59-s − 1.28·61-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: \(0\)
Selberg data: \((2,\ 3640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7612575387\)
\(L(\frac12)\) \(\approx\) \(0.7612575387\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.43T + 3T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + 3.95T + 17T^{2} \)
19 \( 1 + 5.95T + 19T^{2} \)
23 \( 1 + 5.03T + 23T^{2} \)
29 \( 1 - 4.46T + 29T^{2} \)
31 \( 1 - 2.56T + 31T^{2} \)
37 \( 1 - 5.60T + 37T^{2} \)
41 \( 1 + 3.95T + 41T^{2} \)
43 \( 1 + 1.13T + 43T^{2} \)
47 \( 1 - 9.03T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 0.294T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 8.12T + 67T^{2} \)
71 \( 1 - 2.86T + 71T^{2} \)
73 \( 1 + 8.76T + 73T^{2} \)
79 \( 1 - 5.95T + 79T^{2} \)
83 \( 1 - 5.03T + 83T^{2} \)
89 \( 1 - 2.74T + 89T^{2} \)
97 \( 1 + 9.21T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.464108276767221287870821724381, −7.898847017667118859778103230071, −6.89820170844058866871540719525, −6.28566543851752052196195069280, −5.63156341558429639577363175206, −4.57076375070863188575331311390, −4.27971420414855641776744818235, −2.93150573880115084134502196142, −1.98858395828053413216408193115, −0.51233293304449176811288634343, 0.51233293304449176811288634343, 1.98858395828053413216408193115, 2.93150573880115084134502196142, 4.27971420414855641776744818235, 4.57076375070863188575331311390, 5.63156341558429639577363175206, 6.28566543851752052196195069280, 6.89820170844058866871540719525, 7.898847017667118859778103230071, 8.464108276767221287870821724381

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