Properties

Label 2-3640-1.1-c1-0-57
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  − 0.618·3-s + 5-s + 7-s − 2.61·9-s + 2.47·11-s + 13-s − 0.618·15-s − 2.38·17-s − 0.381·19-s − 0.618·21-s − 8.47·23-s + 25-s + 3.47·27-s − 4.61·29-s − 3.09·31-s − 1.52·33-s + 35-s + 7.09·37-s − 0.618·39-s − 6.38·41-s − 5.23·43-s − 2.61·45-s − 4.47·47-s + 49-s + 1.47·51-s + 4.47·53-s + 2.47·55-s + ⋯
L(s)  = 1  − 0.356·3-s + 0.447·5-s + 0.377·7-s − 0.872·9-s + 0.745·11-s + 0.277·13-s − 0.159·15-s − 0.577·17-s − 0.0876·19-s − 0.134·21-s − 1.76·23-s + 0.200·25-s + 0.668·27-s − 0.857·29-s − 0.555·31-s − 0.265·33-s + 0.169·35-s + 1.16·37-s − 0.0989·39-s − 0.996·41-s − 0.798·43-s − 0.390·45-s − 0.652·47-s + 0.142·49-s + 0.206·51-s + 0.614·53-s + 0.333·55-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 \( 1 + 0.618T + 3T^{2} \)
11 \( 1 - 2.47T + 11T^{2} \)
17 \( 1 + 2.38T + 17T^{2} \)
19 \( 1 + 0.381T + 19T^{2} \)
23 \( 1 + 8.47T + 23T^{2} \)
29 \( 1 + 4.61T + 29T^{2} \)
31 \( 1 + 3.09T + 31T^{2} \)
37 \( 1 - 7.09T + 37T^{2} \)
41 \( 1 + 6.38T + 41T^{2} \)
43 \( 1 + 5.23T + 43T^{2} \)
47 \( 1 + 4.47T + 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 0.381T + 67T^{2} \)
71 \( 1 + 15.7T + 71T^{2} \)
73 \( 1 + 6.47T + 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + 8.32T + 89T^{2} \)
97 \( 1 + 2.29T + 97T^{2} \)
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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

−8.307065685524497764506402086081, −7.42914088619605982098954106374, −6.45751529802274101618450950947, −5.99477111280608324298256454866, −5.27588964375858326568990371452, −4.34370917078153160969723114031, −3.53248685223367240810849843416, −2.37512246164586866441784118170, −1.51109463660980875131059846008, 0, 1.51109463660980875131059846008, 2.37512246164586866441784118170, 3.53248685223367240810849843416, 4.34370917078153160969723114031, 5.27588964375858326568990371452, 5.99477111280608324298256454866, 6.45751529802274101618450950947, 7.42914088619605982098954106374, 8.307065685524497764506402086081

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