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Properties

Label 2-2720-680.339-c0-0-3
Degree $2$
Conductor $2720$
Sign $1$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
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 + 13-s − 15-s − 17-s + 19-s + 25-s + 27-s − 29-s + 31-s − 39-s − 47-s + 49-s + 51-s + 53-s − 57-s + 59-s − 61-s + 65-s + 71-s + 73-s − 75-s − 2·79-s − 81-s − 85-s + 87-s − 89-s + ⋯
L(s)  = 1  − 3-s + 5-s + 13-s − 15-s − 17-s + 19-s + 25-s + 27-s − 29-s + 31-s − 39-s − 47-s + 49-s + 51-s + 53-s − 57-s + 59-s − 61-s + 65-s + 71-s + 73-s − 75-s − 2·79-s − 81-s − 85-s + 87-s − 89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $1$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2720} (1359, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.049753143\)
\(L(\frac12)\) \(\approx\) \(1.049753143\)
\(L(1)\) 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 \( 1 - T \)
17 \( 1 + T \)
good3 \( 1 + T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 - T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T + T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T + T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( ( 1 + T )^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T + T^{2} \)
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

−8.995622522117783940440402367558, −8.459888309672847002348147435867, −7.26503202084182913036581662618, −6.49861910375060753609369311332, −5.92124174997527210759013136421, −5.34145684965402451874555086185, −4.51606729476713058672173168413, −3.33392642794848403023264919489, −2.23049463461544830115016484817, −1.04059624861241005661401964404, 1.04059624861241005661401964404, 2.23049463461544830115016484817, 3.33392642794848403023264919489, 4.51606729476713058672173168413, 5.34145684965402451874555086185, 5.92124174997527210759013136421, 6.49861910375060753609369311332, 7.26503202084182913036581662618, 8.459888309672847002348147435867, 8.995622522117783940440402367558

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