Properties

Label 2-3640-1.1-c1-0-52
Degree $2$
Conductor $3640$
Sign $-1$
Analytic cond. $29.0655$
Root an. cond. $5.39124$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 3·9-s + 2·11-s + 13-s + 8·19-s − 8·23-s + 25-s + 6·29-s − 10·31-s + 35-s + 2·37-s + 10·43-s + 3·45-s − 8·47-s + 49-s − 12·53-s − 2·55-s + 6·61-s + 3·63-s − 65-s + 4·71-s − 6·73-s − 2·77-s − 16·79-s + 9·81-s − 4·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 9-s + 0.603·11-s + 0.277·13-s + 1.83·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 1.79·31-s + 0.169·35-s + 0.328·37-s + 1.52·43-s + 0.447·45-s − 1.16·47-s + 1/7·49-s − 1.64·53-s − 0.269·55-s + 0.768·61-s + 0.377·63-s − 0.124·65-s + 0.474·71-s − 0.702·73-s − 0.227·77-s − 1.80·79-s + 81-s − 0.439·83-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: yes
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 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.101917935357030907670183424417, −7.56025189958285642546063172694, −6.64828491987661026865108105537, −5.90351247393730279741713014317, −5.28651403654828433914025679312, −4.16405444916472321316719134794, −3.45627384662986438687428913796, −2.69377150022178809869710718931, −1.35472812286248210433196729500, 0, 1.35472812286248210433196729500, 2.69377150022178809869710718931, 3.45627384662986438687428913796, 4.16405444916472321316719134794, 5.28651403654828433914025679312, 5.90351247393730279741713014317, 6.64828491987661026865108105537, 7.56025189958285642546063172694, 8.101917935357030907670183424417

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