Properties

Label 2-720-1.1-c3-0-22
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 − 20·7-s − 24·11-s + 74·13-s − 54·17-s + 124·19-s − 120·23-s + 25·25-s + 78·29-s − 200·31-s − 100·35-s − 70·37-s − 330·41-s − 92·43-s − 24·47-s + 57·49-s − 450·53-s − 120·55-s + 24·59-s − 322·61-s + 370·65-s + 196·67-s − 288·71-s − 430·73-s + 480·77-s + 520·79-s + 156·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.07·7-s − 0.657·11-s + 1.57·13-s − 0.770·17-s + 1.49·19-s − 1.08·23-s + 1/5·25-s + 0.499·29-s − 1.15·31-s − 0.482·35-s − 0.311·37-s − 1.25·41-s − 0.326·43-s − 0.0744·47-s + 0.166·49-s − 1.16·53-s − 0.294·55-s + 0.0529·59-s − 0.675·61-s + 0.706·65-s + 0.357·67-s − 0.481·71-s − 0.689·73-s + 0.710·77-s + 0.740·79-s + 0.206·83-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: \(1\)
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 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 + 24 T + p^{3} T^{2} \)
13 \( 1 - 74 T + p^{3} T^{2} \)
17 \( 1 + 54 T + p^{3} T^{2} \)
19 \( 1 - 124 T + p^{3} T^{2} \)
23 \( 1 + 120 T + p^{3} T^{2} \)
29 \( 1 - 78 T + p^{3} T^{2} \)
31 \( 1 + 200 T + p^{3} T^{2} \)
37 \( 1 + 70 T + p^{3} T^{2} \)
41 \( 1 + 330 T + p^{3} T^{2} \)
43 \( 1 + 92 T + p^{3} T^{2} \)
47 \( 1 + 24 T + p^{3} T^{2} \)
53 \( 1 + 450 T + p^{3} T^{2} \)
59 \( 1 - 24 T + p^{3} T^{2} \)
61 \( 1 + 322 T + p^{3} T^{2} \)
67 \( 1 - 196 T + p^{3} T^{2} \)
71 \( 1 + 288 T + p^{3} T^{2} \)
73 \( 1 + 430 T + p^{3} T^{2} \)
79 \( 1 - 520 T + p^{3} T^{2} \)
83 \( 1 - 156 T + p^{3} T^{2} \)
89 \( 1 + 1026 T + p^{3} T^{2} \)
97 \( 1 + 286 T + p^{3} T^{2} \)
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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

−9.613143644582392886717763762776, −8.839451752879676223134202541239, −7.897407590603166405793701936968, −6.76700048975469624763125675027, −6.07678816660752157406032692866, −5.20578103779215622029772210758, −3.78942925514281055842750797099, −2.96644104411721720941066105704, −1.55152849596739807232836114335, 0, 1.55152849596739807232836114335, 2.96644104411721720941066105704, 3.78942925514281055842750797099, 5.20578103779215622029772210758, 6.07678816660752157406032692866, 6.76700048975469624763125675027, 7.897407590603166405793701936968, 8.839451752879676223134202541239, 9.613143644582392886717763762776

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