Properties

Label 2-3640-1.1-c1-0-41
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  − 1.21·3-s − 5-s + 7-s − 1.52·9-s − 5.76·11-s − 13-s + 1.21·15-s + 3.42·17-s + 3.23·19-s − 1.21·21-s + 9.23·23-s + 25-s + 5.49·27-s + 6.64·29-s + 1.84·31-s + 7.00·33-s − 35-s + 2.07·37-s + 1.21·39-s − 12.0·41-s − 4.52·43-s + 1.52·45-s + 1.33·47-s + 49-s − 4.16·51-s − 5.05·53-s + 5.76·55-s + ⋯
L(s)  = 1  − 0.702·3-s − 0.447·5-s + 0.377·7-s − 0.507·9-s − 1.73·11-s − 0.277·13-s + 0.313·15-s + 0.830·17-s + 0.743·19-s − 0.265·21-s + 1.92·23-s + 0.200·25-s + 1.05·27-s + 1.23·29-s + 0.330·31-s + 1.21·33-s − 0.169·35-s + 0.340·37-s + 0.194·39-s − 1.88·41-s − 0.690·43-s + 0.226·45-s + 0.194·47-s + 0.142·49-s − 0.582·51-s − 0.694·53-s + 0.776·55-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 \( 1 + 1.21T + 3T^{2} \)
11 \( 1 + 5.76T + 11T^{2} \)
17 \( 1 - 3.42T + 17T^{2} \)
19 \( 1 - 3.23T + 19T^{2} \)
23 \( 1 - 9.23T + 23T^{2} \)
29 \( 1 - 6.64T + 29T^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 - 2.07T + 37T^{2} \)
41 \( 1 + 12.0T + 41T^{2} \)
43 \( 1 + 4.52T + 43T^{2} \)
47 \( 1 - 1.33T + 47T^{2} \)
53 \( 1 + 5.05T + 53T^{2} \)
59 \( 1 + 5.78T + 59T^{2} \)
61 \( 1 + 9.00T + 61T^{2} \)
67 \( 1 + 7.28T + 67T^{2} \)
71 \( 1 - 6.33T + 71T^{2} \)
73 \( 1 + 0.186T + 73T^{2} \)
79 \( 1 + 1.75T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 9.15T + 89T^{2} \)
97 \( 1 + 8.37T + 97T^{2} \)
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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

−8.126976873893232307467757723342, −7.48078378936289933274007398180, −6.73446182281860852496215126690, −5.74134524941966911857121253886, −5.01317339926538442264785952417, −4.80414553659645810738698178801, −3.17474189809578312520743219721, −2.80047521280895651640945899625, −1.18576167267202214895480562939, 0, 1.18576167267202214895480562939, 2.80047521280895651640945899625, 3.17474189809578312520743219721, 4.80414553659645810738698178801, 5.01317339926538442264785952417, 5.74134524941966911857121253886, 6.73446182281860852496215126690, 7.48078378936289933274007398180, 8.126976873893232307467757723342

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