Properties

Label 2-720-1.1-c3-0-8
Degree $2$
Conductor $720$
Sign $1$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s − 14·7-s + 6·11-s + 68·13-s − 78·17-s − 44·19-s + 120·23-s + 25·25-s − 126·29-s + 244·31-s − 70·35-s − 304·37-s + 480·41-s − 104·43-s + 600·47-s − 147·49-s + 258·53-s + 30·55-s + 534·59-s + 362·61-s + 340·65-s + 268·67-s − 972·71-s + 470·73-s − 84·77-s − 1.24e3·79-s + 396·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.164·11-s + 1.45·13-s − 1.11·17-s − 0.531·19-s + 1.08·23-s + 1/5·25-s − 0.806·29-s + 1.41·31-s − 0.338·35-s − 1.35·37-s + 1.82·41-s − 0.368·43-s + 1.86·47-s − 3/7·49-s + 0.668·53-s + 0.0735·55-s + 1.17·59-s + 0.759·61-s + 0.648·65-s + 0.488·67-s − 1.62·71-s + 0.753·73-s − 0.124·77-s − 1.77·79-s + 0.523·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{720} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.060417591\)
\(L(\frac12)\) \(\approx\) \(2.060417591\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
good7 \( 1 + 2 p T + p^{3} T^{2} \)
11 \( 1 - 6 T + p^{3} T^{2} \)
13 \( 1 - 68 T + p^{3} T^{2} \)
17 \( 1 + 78 T + p^{3} T^{2} \)
19 \( 1 + 44 T + p^{3} T^{2} \)
23 \( 1 - 120 T + p^{3} T^{2} \)
29 \( 1 + 126 T + p^{3} T^{2} \)
31 \( 1 - 244 T + p^{3} T^{2} \)
37 \( 1 + 304 T + p^{3} T^{2} \)
41 \( 1 - 480 T + p^{3} T^{2} \)
43 \( 1 + 104 T + p^{3} T^{2} \)
47 \( 1 - 600 T + p^{3} T^{2} \)
53 \( 1 - 258 T + p^{3} T^{2} \)
59 \( 1 - 534 T + p^{3} T^{2} \)
61 \( 1 - 362 T + p^{3} T^{2} \)
67 \( 1 - 4 p T + p^{3} T^{2} \)
71 \( 1 + 972 T + p^{3} T^{2} \)
73 \( 1 - 470 T + p^{3} T^{2} \)
79 \( 1 + 1244 T + p^{3} T^{2} \)
83 \( 1 - 396 T + p^{3} T^{2} \)
89 \( 1 - 972 T + p^{3} T^{2} \)
97 \( 1 + 46 T + p^{3} T^{2} \)
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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

−10.05382074208939394635026393311, −8.964420673411454380233860357584, −8.655409251655525632856860891299, −7.20777975797714649638427767759, −6.42493483199127401119070966294, −5.74237118502087885495443180463, −4.43470751510138670131690785994, −3.43621740286061201520810999956, −2.24806435788978748436312181830, −0.832184243272800846755151944932, 0.832184243272800846755151944932, 2.24806435788978748436312181830, 3.43621740286061201520810999956, 4.43470751510138670131690785994, 5.74237118502087885495443180463, 6.42493483199127401119070966294, 7.20777975797714649638427767759, 8.655409251655525632856860891299, 8.964420673411454380233860357584, 10.05382074208939394635026393311

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