Properties

Label 2-6720-1.1-c1-0-21
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 − 6.47·11-s − 4.47·13-s + 15-s − 2·17-s + 2.47·19-s + 21-s + 4·23-s + 25-s + 27-s + 2·29-s + 1.52·31-s − 6.47·33-s + 35-s + 6.94·37-s − 4.47·39-s − 2·41-s − 8.94·43-s + 45-s + 12.9·47-s + 49-s − 2·51-s + 3.52·53-s − 6.47·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s − 1.95·11-s − 1.24·13-s + 0.258·15-s − 0.485·17-s + 0.567·19-s + 0.218·21-s + 0.834·23-s + 0.200·25-s + 0.192·27-s + 0.371·29-s + 0.274·31-s − 1.12·33-s + 0.169·35-s + 1.14·37-s − 0.716·39-s − 0.312·41-s − 1.36·43-s + 0.149·45-s + 1.88·47-s + 0.142·49-s − 0.280·51-s + 0.484·53-s − 0.872·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6720,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.320310399\)
\(L(\frac12)\) \(\approx\) \(2.320310399\)
\(L(\frac{3}{2})\) 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 \( 1 + 6.47T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 2.47T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 1.52T + 31T^{2} \)
37 \( 1 - 6.94T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 8.94T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 - 3.52T + 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 5.52T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 8.47T + 97T^{2} \)
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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

−7.889478134736381045378421684149, −7.45061846569692521513440648924, −6.76522196805702942445661065025, −5.69577936613152498189656276558, −5.02716554460061928771695414340, −4.64131812355871528825673077112, −3.38486087557075823720941675107, −2.50841151368139800649541777091, −2.22011211674731068404610321639, −0.73362701042985454210333249150, 0.73362701042985454210333249150, 2.22011211674731068404610321639, 2.50841151368139800649541777091, 3.38486087557075823720941675107, 4.64131812355871528825673077112, 5.02716554460061928771695414340, 5.69577936613152498189656276558, 6.76522196805702942445661065025, 7.45061846569692521513440648924, 7.889478134736381045378421684149

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