Properties

Label 2-720-1.1-c3-0-13
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 + 24·7-s + 52·11-s + 22·13-s + 14·17-s + 20·19-s − 168·23-s + 25·25-s − 230·29-s + 288·31-s − 120·35-s − 34·37-s − 122·41-s + 188·43-s + 256·47-s + 233·49-s + 338·53-s − 260·55-s + 100·59-s + 742·61-s − 110·65-s + 84·67-s − 328·71-s − 38·73-s + 1.24e3·77-s + 240·79-s + 1.21e3·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.29·7-s + 1.42·11-s + 0.469·13-s + 0.199·17-s + 0.241·19-s − 1.52·23-s + 1/5·25-s − 1.47·29-s + 1.66·31-s − 0.579·35-s − 0.151·37-s − 0.464·41-s + 0.666·43-s + 0.794·47-s + 0.679·49-s + 0.875·53-s − 0.637·55-s + 0.220·59-s + 1.55·61-s − 0.209·65-s + 0.153·67-s − 0.548·71-s − 0.0609·73-s + 1.84·77-s + 0.341·79-s + 1.60·83-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

\(L(2)\) \(\approx\) \(2.466945721\)
\(L(\frac12)\) \(\approx\) \(2.466945721\)
\(L(\frac{5}{2})\) 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 \( 1 - 24 T + p^{3} T^{2} \)
11 \( 1 - 52 T + p^{3} T^{2} \)
13 \( 1 - 22 T + p^{3} T^{2} \)
17 \( 1 - 14 T + p^{3} T^{2} \)
19 \( 1 - 20 T + p^{3} T^{2} \)
23 \( 1 + 168 T + p^{3} T^{2} \)
29 \( 1 + 230 T + p^{3} T^{2} \)
31 \( 1 - 288 T + p^{3} T^{2} \)
37 \( 1 + 34 T + p^{3} T^{2} \)
41 \( 1 + 122 T + p^{3} T^{2} \)
43 \( 1 - 188 T + p^{3} T^{2} \)
47 \( 1 - 256 T + p^{3} T^{2} \)
53 \( 1 - 338 T + p^{3} T^{2} \)
59 \( 1 - 100 T + p^{3} T^{2} \)
61 \( 1 - 742 T + p^{3} T^{2} \)
67 \( 1 - 84 T + p^{3} T^{2} \)
71 \( 1 + 328 T + p^{3} T^{2} \)
73 \( 1 + 38 T + p^{3} T^{2} \)
79 \( 1 - 240 T + p^{3} T^{2} \)
83 \( 1 - 1212 T + p^{3} T^{2} \)
89 \( 1 + 330 T + p^{3} T^{2} \)
97 \( 1 - 866 T + p^{3} T^{2} \)
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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

−10.03592655644087435448539811614, −9.023850903589782693711853777871, −8.262917398673117374620972709533, −7.55547112557296945683443887016, −6.49471603265344704319544748879, −5.51254354899710529348763514242, −4.35064539347210923658634660782, −3.70989584895754066074388930148, −2.03007275694702051065720781230, −0.965873354858502213873430288091, 0.965873354858502213873430288091, 2.03007275694702051065720781230, 3.70989584895754066074388930148, 4.35064539347210923658634660782, 5.51254354899710529348763514242, 6.49471603265344704319544748879, 7.55547112557296945683443887016, 8.262917398673117374620972709533, 9.023850903589782693711853777871, 10.03592655644087435448539811614

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