Properties

Label 2-3640-1.1-c1-0-6
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  − 2.68·3-s − 5-s + 7-s + 4.22·9-s − 13-s + 2.68·15-s + 1.22·17-s − 0.777·19-s − 2.68·21-s + 7.81·23-s + 25-s − 3.28·27-s − 7.13·29-s + 1.31·31-s − 35-s − 8.50·37-s + 2.68·39-s + 1.22·41-s + 1.37·43-s − 4.22·45-s − 3.81·47-s + 49-s − 3.28·51-s + 2·53-s + 2.09·57-s − 4.06·59-s − 10·61-s + ⋯
L(s)  = 1  − 1.55·3-s − 0.447·5-s + 0.377·7-s + 1.40·9-s − 0.277·13-s + 0.693·15-s + 0.296·17-s − 0.178·19-s − 0.586·21-s + 1.63·23-s + 0.200·25-s − 0.632·27-s − 1.32·29-s + 0.235·31-s − 0.169·35-s − 1.39·37-s + 0.430·39-s + 0.190·41-s + 0.209·43-s − 0.629·45-s − 0.557·47-s + 0.142·49-s − 0.459·51-s + 0.274·53-s + 0.276·57-s − 0.528·59-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: $\chi_{3640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8006221169\)
\(L(\frac12)\) \(\approx\) \(0.8006221169\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.68T + 3T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 1.22T + 17T^{2} \)
19 \( 1 + 0.777T + 19T^{2} \)
23 \( 1 - 7.81T + 23T^{2} \)
29 \( 1 + 7.13T + 29T^{2} \)
31 \( 1 - 1.31T + 31T^{2} \)
37 \( 1 + 8.50T + 37T^{2} \)
41 \( 1 - 1.22T + 41T^{2} \)
43 \( 1 - 1.37T + 43T^{2} \)
47 \( 1 + 3.81T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 4.06T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 5.37T + 71T^{2} \)
73 \( 1 + 0.930T + 73T^{2} \)
79 \( 1 - 0.777T + 79T^{2} \)
83 \( 1 + 7.81T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 - 19.0T + 97T^{2} \)
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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

−8.524292984173163514023410396298, −7.54312165642481378372799501941, −7.02563063596333029396700723542, −6.27671680147623795093464811242, −5.38030660287290178254403216167, −4.99364018922852794547145236527, −4.14134494086035703470492591008, −3.11323311213912354113152463034, −1.68587159636540516011195790087, −0.58017825424418269792892065228, 0.58017825424418269792892065228, 1.68587159636540516011195790087, 3.11323311213912354113152463034, 4.14134494086035703470492591008, 4.99364018922852794547145236527, 5.38030660287290178254403216167, 6.27671680147623795093464811242, 7.02563063596333029396700723542, 7.54312165642481378372799501941, 8.524292984173163514023410396298

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