Properties

Label 2-2240-35.34-c0-0-3
Degree $2$
Conductor $2240$
Sign $1$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 7-s + 11-s + 13-s − 15-s − 17-s + 21-s + 25-s − 27-s + 29-s + 33-s − 35-s + 39-s − 47-s + 49-s − 51-s − 55-s − 65-s + 2·71-s + 2·73-s + 75-s + 77-s − 79-s − 81-s − 2·83-s + 85-s + ⋯
L(s)  = 1  + 3-s − 5-s + 7-s + 11-s + 13-s − 15-s − 17-s + 21-s + 25-s − 27-s + 29-s + 33-s − 35-s + 39-s − 47-s + 49-s − 51-s − 55-s − 65-s + 2·71-s + 2·73-s + 75-s + 77-s − 79-s − 81-s − 2·83-s + 85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2240} (769, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.609556686\)
\(L(\frac12)\) \(\approx\) \(1.609556686\)
\(L(1)\) 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 \( 1 - T \)
good3 \( 1 - T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 - T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )^{2} \)
73 \( ( 1 - T )^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 + T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−8.820738877963536642368942996802, −8.500227427684979222680892116017, −7.974148538596621093813012882005, −7.04287686447460118284552543387, −6.28106336659729637455684868216, −5.01962788217033758359746052772, −4.14183443191368691548019728093, −3.59232776293508694724433552002, −2.51374964405774669040363009954, −1.33514658438003097784799390782, 1.33514658438003097784799390782, 2.51374964405774669040363009954, 3.59232776293508694724433552002, 4.14183443191368691548019728093, 5.01962788217033758359746052772, 6.28106336659729637455684868216, 7.04287686447460118284552543387, 7.974148538596621093813012882005, 8.500227427684979222680892116017, 8.820738877963536642368942996802

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