Properties

Label 2-3640-1.1-c1-0-63
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.32·3-s − 5-s + 7-s − 1.24·9-s + 2.54·11-s − 13-s − 1.32·15-s − 3.92·17-s − 4.79·19-s + 1.32·21-s − 4.70·23-s + 25-s − 5.62·27-s + 4.32·29-s + 8.37·31-s + 3.37·33-s − 35-s − 1.69·37-s − 1.32·39-s + 4.84·41-s − 11.8·43-s + 1.24·45-s − 10.8·47-s + 49-s − 5.19·51-s − 9.04·53-s − 2.54·55-s + ⋯
L(s)  = 1  + 0.764·3-s − 0.447·5-s + 0.377·7-s − 0.415·9-s + 0.767·11-s − 0.277·13-s − 0.341·15-s − 0.952·17-s − 1.09·19-s + 0.288·21-s − 0.980·23-s + 0.200·25-s − 1.08·27-s + 0.802·29-s + 1.50·31-s + 0.586·33-s − 0.169·35-s − 0.279·37-s − 0.212·39-s + 0.756·41-s − 1.80·43-s + 0.185·45-s − 1.57·47-s + 0.142·49-s − 0.727·51-s − 1.24·53-s − 0.343·55-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: $\chi_{3640} (1, \cdot )$
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 \( 1 - 1.32T + 3T^{2} \)
11 \( 1 - 2.54T + 11T^{2} \)
17 \( 1 + 3.92T + 17T^{2} \)
19 \( 1 + 4.79T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 - 4.32T + 29T^{2} \)
31 \( 1 - 8.37T + 31T^{2} \)
37 \( 1 + 1.69T + 37T^{2} \)
41 \( 1 - 4.84T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 9.04T + 53T^{2} \)
59 \( 1 + 4.69T + 59T^{2} \)
61 \( 1 + 5.37T + 61T^{2} \)
67 \( 1 - 6.59T + 67T^{2} \)
71 \( 1 + 0.527T + 71T^{2} \)
73 \( 1 - 4.27T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 - 1.95T + 83T^{2} \)
89 \( 1 + 7.91T + 89T^{2} \)
97 \( 1 - 4.32T + 97T^{2} \)
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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

−8.428311843543538615260333159330, −7.65539807991943344084277556119, −6.59220554722537172615594067714, −6.20349999653932809990768381204, −4.86002039266486887074775415955, −4.31430885990216571924302405343, −3.42164879884499011076632028910, −2.54418167843084901512021830060, −1.65066095981150055762927947751, 0, 1.65066095981150055762927947751, 2.54418167843084901512021830060, 3.42164879884499011076632028910, 4.31430885990216571924302405343, 4.86002039266486887074775415955, 6.20349999653932809990768381204, 6.59220554722537172615594067714, 7.65539807991943344084277556119, 8.428311843543538615260333159330

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