Properties

Label 2-720-1.1-c1-0-2
Degree $2$
Conductor $720$
Sign $1$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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  + 5-s − 2·7-s + 2·13-s + 6·17-s + 4·19-s + 6·23-s + 25-s − 6·29-s + 4·31-s − 2·35-s + 2·37-s − 6·41-s + 10·43-s − 6·47-s − 3·49-s + 6·53-s + 12·59-s + 2·61-s + 2·65-s − 2·67-s − 12·71-s + 2·73-s − 8·79-s + 6·83-s + 6·85-s + 6·89-s − 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.338·35-s + 0.328·37-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s + 0.824·53-s + 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.244·67-s − 1.42·71-s + 0.234·73-s − 0.900·79-s + 0.658·83-s + 0.650·85-s + 0.635·89-s − 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.630653748\)
\(L(\frac12)\) \(\approx\) \(1.630653748\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−10.22754300302114228185381817663, −9.633261704500415779063377753491, −8.847850276232423443211930888771, −7.74818737522785607031159368119, −6.88656092970954199609923322632, −5.90138678053477774939394585997, −5.17606419420266207753384183426, −3.70610300459195515417889608627, −2.84705552122166969163585152299, −1.16915866227454488376692012454, 1.16915866227454488376692012454, 2.84705552122166969163585152299, 3.70610300459195515417889608627, 5.17606419420266207753384183426, 5.90138678053477774939394585997, 6.88656092970954199609923322632, 7.74818737522785607031159368119, 8.847850276232423443211930888771, 9.633261704500415779063377753491, 10.22754300302114228185381817663

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