Properties

Label 2-2240-1.1-c1-0-32
Degree $2$
Conductor $2240$
Sign $1$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 5-s + 7-s + 6·9-s + 5·11-s + 5·13-s − 3·15-s − 7·17-s + 2·19-s + 3·21-s − 2·23-s + 25-s + 9·27-s − 7·29-s + 4·31-s + 15·33-s − 35-s + 6·37-s + 15·39-s − 12·41-s + 2·43-s − 6·45-s + 47-s + 49-s − 21·51-s − 5·55-s + 6·57-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s + 0.377·7-s + 2·9-s + 1.50·11-s + 1.38·13-s − 0.774·15-s − 1.69·17-s + 0.458·19-s + 0.654·21-s − 0.417·23-s + 1/5·25-s + 1.73·27-s − 1.29·29-s + 0.718·31-s + 2.61·33-s − 0.169·35-s + 0.986·37-s + 2.40·39-s − 1.87·41-s + 0.304·43-s − 0.894·45-s + 0.145·47-s + 1/7·49-s − 2.94·51-s − 0.674·55-s + 0.794·57-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.667490059\)
\(L(\frac12)\) \(\approx\) \(3.667490059\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 - p T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−8.929328035580339063333330502851, −8.442047347536339396857622970932, −7.70322279184072217617891578257, −6.87040510040780692294024001121, −6.16504537306361424428061030060, −4.60306475102241152447956391011, −3.91397125713061338961016197906, −3.41098805715618011663284982660, −2.18719924749037536985676815986, −1.33612315338884046761653874128, 1.33612315338884046761653874128, 2.18719924749037536985676815986, 3.41098805715618011663284982660, 3.91397125713061338961016197906, 4.60306475102241152447956391011, 6.16504537306361424428061030060, 6.87040510040780692294024001121, 7.70322279184072217617891578257, 8.442047347536339396857622970932, 8.929328035580339063333330502851

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