Properties

Label 2-720-1.1-c3-0-5
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 + 4·7-s + 12·11-s − 58·13-s − 66·17-s + 100·19-s + 132·23-s + 25·25-s + 90·29-s − 152·31-s − 20·35-s − 34·37-s + 438·41-s − 32·43-s − 204·47-s − 327·49-s − 222·53-s − 60·55-s + 420·59-s + 902·61-s + 290·65-s + 1.02e3·67-s + 432·71-s + 362·73-s + 48·77-s + 160·79-s + 72·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.215·7-s + 0.328·11-s − 1.23·13-s − 0.941·17-s + 1.20·19-s + 1.19·23-s + 1/5·25-s + 0.576·29-s − 0.880·31-s − 0.0965·35-s − 0.151·37-s + 1.66·41-s − 0.113·43-s − 0.633·47-s − 0.953·49-s − 0.575·53-s − 0.147·55-s + 0.926·59-s + 1.89·61-s + 0.553·65-s + 1.86·67-s + 0.722·71-s + 0.580·73-s + 0.0710·77-s + 0.227·79-s + 0.0952·83-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

\(L(2)\) \(\approx\) \(1.698266296\)
\(L(\frac12)\) \(\approx\) \(1.698266296\)
\(L(\frac{5}{2})\) 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 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 - 12 T + p^{3} T^{2} \)
13 \( 1 + 58 T + p^{3} T^{2} \)
17 \( 1 + 66 T + p^{3} T^{2} \)
19 \( 1 - 100 T + p^{3} T^{2} \)
23 \( 1 - 132 T + p^{3} T^{2} \)
29 \( 1 - 90 T + p^{3} T^{2} \)
31 \( 1 + 152 T + p^{3} T^{2} \)
37 \( 1 + 34 T + p^{3} T^{2} \)
41 \( 1 - 438 T + p^{3} T^{2} \)
43 \( 1 + 32 T + p^{3} T^{2} \)
47 \( 1 + 204 T + p^{3} T^{2} \)
53 \( 1 + 222 T + p^{3} T^{2} \)
59 \( 1 - 420 T + p^{3} T^{2} \)
61 \( 1 - 902 T + p^{3} T^{2} \)
67 \( 1 - 1024 T + p^{3} T^{2} \)
71 \( 1 - 432 T + p^{3} T^{2} \)
73 \( 1 - 362 T + p^{3} T^{2} \)
79 \( 1 - 160 T + p^{3} T^{2} \)
83 \( 1 - 72 T + p^{3} T^{2} \)
89 \( 1 + 810 T + p^{3} T^{2} \)
97 \( 1 - 1106 T + p^{3} T^{2} \)
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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

−9.872978703856461894043015861411, −9.239395075058080752134280715308, −8.258302383397885714975232606182, −7.33609288230875640139932095238, −6.71431311286194958059600056474, −5.32884233645066119963398392418, −4.61588179213245866675192192296, −3.43183326484512113881131974588, −2.27736882290125384233412313843, −0.74322804685502860806979532349, 0.74322804685502860806979532349, 2.27736882290125384233412313843, 3.43183326484512113881131974588, 4.61588179213245866675192192296, 5.32884233645066119963398392418, 6.71431311286194958059600056474, 7.33609288230875640139932095238, 8.258302383397885714975232606182, 9.239395075058080752134280715308, 9.872978703856461894043015861411

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