Properties

Label 2-720-1.1-c3-0-21
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·5-s + 4·11-s + 54·13-s − 114·17-s − 44·19-s + 96·23-s + 25·25-s − 134·29-s + 272·31-s − 98·37-s + 6·41-s − 12·43-s − 200·47-s − 343·49-s − 654·53-s − 20·55-s + 36·59-s − 442·61-s − 270·65-s + 188·67-s − 632·71-s − 390·73-s − 688·79-s + 1.18e3·83-s + 570·85-s + 694·89-s + 220·95-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.109·11-s + 1.15·13-s − 1.62·17-s − 0.531·19-s + 0.870·23-s + 1/5·25-s − 0.858·29-s + 1.57·31-s − 0.435·37-s + 0.0228·41-s − 0.0425·43-s − 0.620·47-s − 49-s − 1.69·53-s − 0.0490·55-s + 0.0794·59-s − 0.927·61-s − 0.515·65-s + 0.342·67-s − 1.05·71-s − 0.625·73-s − 0.979·79-s + 1.57·83-s + 0.727·85-s + 0.826·89-s + 0.237·95-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: \(1\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p^{3} T^{2} \)
11 \( 1 - 4 T + p^{3} T^{2} \)
13 \( 1 - 54 T + p^{3} T^{2} \)
17 \( 1 + 114 T + p^{3} T^{2} \)
19 \( 1 + 44 T + p^{3} T^{2} \)
23 \( 1 - 96 T + p^{3} T^{2} \)
29 \( 1 + 134 T + p^{3} T^{2} \)
31 \( 1 - 272 T + p^{3} T^{2} \)
37 \( 1 + 98 T + p^{3} T^{2} \)
41 \( 1 - 6 T + p^{3} T^{2} \)
43 \( 1 + 12 T + p^{3} T^{2} \)
47 \( 1 + 200 T + p^{3} T^{2} \)
53 \( 1 + 654 T + p^{3} T^{2} \)
59 \( 1 - 36 T + p^{3} T^{2} \)
61 \( 1 + 442 T + p^{3} T^{2} \)
67 \( 1 - 188 T + p^{3} T^{2} \)
71 \( 1 + 632 T + p^{3} T^{2} \)
73 \( 1 + 390 T + p^{3} T^{2} \)
79 \( 1 + 688 T + p^{3} T^{2} \)
83 \( 1 - 1188 T + p^{3} T^{2} \)
89 \( 1 - 694 T + p^{3} T^{2} \)
97 \( 1 + 1726 T + p^{3} T^{2} \)
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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

−9.441592853999042607847323646935, −8.691636308798170730205431417433, −7.995268274334997955762032999023, −6.80148373918685823544763409531, −6.22288113544855041840252182719, −4.87083806250021007745933914505, −4.04222466359204504078249596149, −2.92112734152836684564415836551, −1.50886266292918633478968469462, 0, 1.50886266292918633478968469462, 2.92112734152836684564415836551, 4.04222466359204504078249596149, 4.87083806250021007745933914505, 6.22288113544855041840252182719, 6.80148373918685823544763409531, 7.995268274334997955762032999023, 8.691636308798170730205431417433, 9.441592853999042607847323646935

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