Properties

Label 2-6720-1.1-c1-0-60
Degree $2$
Conductor $6720$
Sign $-1$
Analytic cond. $53.6594$
Root an. cond. $7.32526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 7-s + 9-s − 4.47·13-s + 15-s + 4.47·17-s − 21-s − 2.47·23-s + 25-s − 27-s + 0.472·29-s + 2.47·31-s − 35-s + 0.472·37-s + 4.47·39-s − 6.94·41-s − 1.52·43-s − 45-s + 6.47·47-s + 49-s − 4.47·51-s − 2·53-s + 4·59-s − 3.52·61-s + 63-s + 4.47·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s − 1.24·13-s + 0.258·15-s + 1.08·17-s − 0.218·21-s − 0.515·23-s + 0.200·25-s − 0.192·27-s + 0.0876·29-s + 0.444·31-s − 0.169·35-s + 0.0776·37-s + 0.716·39-s − 1.08·41-s − 0.232·43-s − 0.149·45-s + 0.944·47-s + 0.142·49-s − 0.626·51-s − 0.274·53-s + 0.520·59-s − 0.451·61-s + 0.125·63-s + 0.554·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6720\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(53.6594\)
Root analytic conductor: \(7.32526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6720} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6720,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 - 0.472T + 29T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 - 0.472T + 37T^{2} \)
41 \( 1 + 6.94T + 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 - 6.47T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 3.52T + 61T^{2} \)
67 \( 1 - 6.47T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 0.472T + 73T^{2} \)
79 \( 1 + 4.94T + 79T^{2} \)
83 \( 1 + 0.944T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 4.47T + 97T^{2} \)
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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

−7.58938889458375342114894915733, −7.04067241544700805494926835356, −6.19601871327697399192661268510, −5.40197774647185895400154717762, −4.85546647127553761811008252183, −4.11736709701848401100996505607, −3.22105032619658078629174650559, −2.25594340468474005481396056674, −1.16515863644149473107174368034, 0, 1.16515863644149473107174368034, 2.25594340468474005481396056674, 3.22105032619658078629174650559, 4.11736709701848401100996505607, 4.85546647127553761811008252183, 5.40197774647185895400154717762, 6.19601871327697399192661268510, 7.04067241544700805494926835356, 7.58938889458375342114894915733

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