Properties

Label 2-720-1.1-c3-0-10
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 + 18·7-s + 34·11-s + 12·13-s + 102·17-s − 164·19-s + 48·23-s + 25·25-s − 146·29-s − 100·31-s − 90·35-s + 328·37-s + 288·41-s − 120·43-s + 16·47-s − 19·49-s + 126·53-s − 170·55-s + 642·59-s + 602·61-s − 60·65-s − 436·67-s + 652·71-s + 1.06e3·73-s + 612·77-s − 388·79-s − 444·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.971·7-s + 0.931·11-s + 0.256·13-s + 1.45·17-s − 1.98·19-s + 0.435·23-s + 1/5·25-s − 0.934·29-s − 0.579·31-s − 0.434·35-s + 1.45·37-s + 1.09·41-s − 0.425·43-s + 0.0496·47-s − 0.0553·49-s + 0.326·53-s − 0.416·55-s + 1.41·59-s + 1.26·61-s − 0.114·65-s − 0.795·67-s + 1.08·71-s + 1.70·73-s + 0.905·77-s − 0.552·79-s − 0.587·83-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

\(L(2)\) \(\approx\) \(2.251488299\)
\(L(\frac12)\) \(\approx\) \(2.251488299\)
\(L(\frac{5}{2})\) 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 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 - 34 T + p^{3} T^{2} \)
13 \( 1 - 12 T + p^{3} T^{2} \)
17 \( 1 - 6 p T + p^{3} T^{2} \)
19 \( 1 + 164 T + p^{3} T^{2} \)
23 \( 1 - 48 T + p^{3} T^{2} \)
29 \( 1 + 146 T + p^{3} T^{2} \)
31 \( 1 + 100 T + p^{3} T^{2} \)
37 \( 1 - 328 T + p^{3} T^{2} \)
41 \( 1 - 288 T + p^{3} T^{2} \)
43 \( 1 + 120 T + p^{3} T^{2} \)
47 \( 1 - 16 T + p^{3} T^{2} \)
53 \( 1 - 126 T + p^{3} T^{2} \)
59 \( 1 - 642 T + p^{3} T^{2} \)
61 \( 1 - 602 T + p^{3} T^{2} \)
67 \( 1 + 436 T + p^{3} T^{2} \)
71 \( 1 - 652 T + p^{3} T^{2} \)
73 \( 1 - 1062 T + p^{3} T^{2} \)
79 \( 1 + 388 T + p^{3} T^{2} \)
83 \( 1 + 444 T + p^{3} T^{2} \)
89 \( 1 - 820 T + p^{3} T^{2} \)
97 \( 1 + 766 T + p^{3} T^{2} \)
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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

−10.04534780127088484827850303380, −9.032435453030411864056009700848, −8.262902253591046103857758184537, −7.54256638130352250611812336959, −6.49801555556306011329787617466, −5.51068309476131649156254100906, −4.39413722704967988010938954747, −3.64372257758691318194324532568, −2.09744370762337834890481466836, −0.900911889672021559446328700061, 0.900911889672021559446328700061, 2.09744370762337834890481466836, 3.64372257758691318194324532568, 4.39413722704967988010938954747, 5.51068309476131649156254100906, 6.49801555556306011329787617466, 7.54256638130352250611812336959, 8.262902253591046103857758184537, 9.032435453030411864056009700848, 10.04534780127088484827850303380

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