Properties

Label 2-720-1.1-c3-0-0
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 − 32·7-s − 60·11-s − 34·13-s − 42·17-s + 76·19-s + 25·25-s − 6·29-s + 232·31-s + 160·35-s + 134·37-s − 234·41-s + 412·43-s − 360·47-s + 681·49-s − 222·53-s + 300·55-s + 660·59-s − 490·61-s + 170·65-s − 812·67-s + 120·71-s + 746·73-s + 1.92e3·77-s − 152·79-s − 804·83-s + 210·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.72·7-s − 1.64·11-s − 0.725·13-s − 0.599·17-s + 0.917·19-s + 1/5·25-s − 0.0384·29-s + 1.34·31-s + 0.772·35-s + 0.595·37-s − 0.891·41-s + 1.46·43-s − 1.11·47-s + 1.98·49-s − 0.575·53-s + 0.735·55-s + 1.45·59-s − 1.02·61-s + 0.324·65-s − 1.48·67-s + 0.200·71-s + 1.19·73-s + 2.84·77-s − 0.216·79-s − 1.06·83-s + 0.267·85-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

\(L(2)\) \(\approx\) \(0.6721812809\)
\(L(\frac12)\) \(\approx\) \(0.6721812809\)
\(L(\frac{5}{2})\) 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 \( 1 + 32 T + p^{3} T^{2} \)
11 \( 1 + 60 T + p^{3} T^{2} \)
13 \( 1 + 34 T + p^{3} T^{2} \)
17 \( 1 + 42 T + p^{3} T^{2} \)
19 \( 1 - 4 p T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + 6 T + p^{3} T^{2} \)
31 \( 1 - 232 T + p^{3} T^{2} \)
37 \( 1 - 134 T + p^{3} T^{2} \)
41 \( 1 + 234 T + p^{3} T^{2} \)
43 \( 1 - 412 T + p^{3} T^{2} \)
47 \( 1 + 360 T + p^{3} T^{2} \)
53 \( 1 + 222 T + p^{3} T^{2} \)
59 \( 1 - 660 T + p^{3} T^{2} \)
61 \( 1 + 490 T + p^{3} T^{2} \)
67 \( 1 + 812 T + p^{3} T^{2} \)
71 \( 1 - 120 T + p^{3} T^{2} \)
73 \( 1 - 746 T + p^{3} T^{2} \)
79 \( 1 + 152 T + p^{3} T^{2} \)
83 \( 1 + 804 T + p^{3} T^{2} \)
89 \( 1 - 678 T + p^{3} T^{2} \)
97 \( 1 - 2 p T + p^{3} T^{2} \)
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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

−9.972818579620927329261514313836, −9.358893224855123418628068909529, −8.190629843711450829484287535703, −7.38631396398596392401750419690, −6.56090764603766315595300063513, −5.55408719688874687030267058323, −4.52627400135944997683761290361, −3.20985699988465287320024599894, −2.58703497053564748485326602717, −0.43788944964121694309058618898, 0.43788944964121694309058618898, 2.58703497053564748485326602717, 3.20985699988465287320024599894, 4.52627400135944997683761290361, 5.55408719688874687030267058323, 6.56090764603766315595300063513, 7.38631396398596392401750419690, 8.190629843711450829484287535703, 9.358893224855123418628068909529, 9.972818579620927329261514313836

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