Properties

Label 2-6720-1.1-c1-0-54
Degree $2$
Conductor $6720$
Sign $-1$
Analytic cond. $53.6594$
Root an. cond. $7.32526$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 4·11-s + 2·13-s + 15-s + 2·17-s − 4·19-s + 21-s + 25-s − 27-s + 2·29-s + 8·31-s + 4·33-s + 35-s + 2·37-s − 2·39-s + 2·41-s − 4·43-s − 45-s + 49-s − 2·51-s + 10·53-s + 4·55-s + 4·57-s + 12·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.169·35-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s + 1.37·53-s + 0.539·55-s + 0.529·57-s + 1.56·59-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: yes
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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−7.61465113149376131388775328245, −6.90940038220234233328140871933, −6.16458557561649791843039843687, −5.59613722468513539505922609511, −4.74222449423502780904305824189, −4.11664567523245606800028850898, −3.15828342333522043788763466295, −2.38873475787086144556332957685, −1.06885418828923509331016753538, 0, 1.06885418828923509331016753538, 2.38873475787086144556332957685, 3.15828342333522043788763466295, 4.11664567523245606800028850898, 4.74222449423502780904305824189, 5.59613722468513539505922609511, 6.16458557561649791843039843687, 6.90940038220234233328140871933, 7.61465113149376131388775328245

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