Properties

Label 2-6720-1.1-c1-0-16
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 7-s + 9-s + 3.12·11-s − 5.12·13-s − 15-s + 2·17-s − 7.12·19-s − 21-s + 25-s − 27-s + 2·29-s − 3.12·31-s − 3.12·33-s + 35-s + 2·37-s + 5.12·39-s + 2·41-s + 6.24·43-s + 45-s + 49-s − 2·51-s + 11.3·53-s + 3.12·55-s + 7.12·57-s + 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.941·11-s − 1.42·13-s − 0.258·15-s + 0.485·17-s − 1.63·19-s − 0.218·21-s + 0.200·25-s − 0.192·27-s + 0.371·29-s − 0.560·31-s − 0.543·33-s + 0.169·35-s + 0.328·37-s + 0.820·39-s + 0.312·41-s + 0.952·43-s + 0.149·45-s + 0.142·49-s − 0.280·51-s + 1.56·53-s + 0.421·55-s + 0.943·57-s + 0.520·59-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: no
Arithmetic: yes
Character: $\chi_{6720} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6720,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.688502222\)
\(L(\frac12)\) \(\approx\) \(1.688502222\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.12T + 11T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 3.12T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 6.24T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 8.24T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 - 3.12T + 71T^{2} \)
73 \( 1 - 6.87T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−7.82389758625058590301950783520, −7.25164144335442151100501031169, −6.46946987729039197030577977881, −5.96153218349929408049912422304, −5.09880520705935399400457081385, −4.51102712715630081017310728516, −3.77147443368869694533950532494, −2.52273585610845977470953891964, −1.84208102599593655266593948468, −0.68727139888052538684776912316, 0.68727139888052538684776912316, 1.84208102599593655266593948468, 2.52273585610845977470953891964, 3.77147443368869694533950532494, 4.51102712715630081017310728516, 5.09880520705935399400457081385, 5.96153218349929408049912422304, 6.46946987729039197030577977881, 7.25164144335442151100501031169, 7.82389758625058590301950783520

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