Properties

Label 2-1680-1.1-c1-0-21
Degree $2$
Conductor $1680$
Sign $-1$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
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 + 2.47·11-s − 4.47·13-s − 15-s − 2·17-s − 6.47·19-s − 21-s − 4·23-s + 25-s + 27-s − 2·29-s − 10.4·31-s + 2.47·33-s + 35-s + 10.9·37-s − 4.47·39-s − 2·41-s + 8.94·43-s − 45-s + 4.94·47-s + 49-s − 2·51-s − 12.4·53-s − 2.47·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.745·11-s − 1.24·13-s − 0.258·15-s − 0.485·17-s − 1.48·19-s − 0.218·21-s − 0.834·23-s + 0.200·25-s + 0.192·27-s − 0.371·29-s − 1.88·31-s + 0.430·33-s + 0.169·35-s + 1.79·37-s − 0.716·39-s − 0.312·41-s + 1.36·43-s − 0.149·45-s + 0.721·47-s + 0.142·49-s − 0.280·51-s − 1.71·53-s − 0.333·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ -1)\)

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 \( 1 - 2.47T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 6.47T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 - 4.94T + 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + 3.52T + 73T^{2} \)
79 \( 1 - 4.94T + 79T^{2} \)
83 \( 1 + 0.944T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 0.472T + 97T^{2} \)
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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

−9.201657130364874607788912355154, −8.095877094651885857593761101144, −7.46283883012578601583213147088, −6.66323789863247983935396491695, −5.80989038895996252692901571045, −4.46143013051310227158833590017, −4.00522230305247131920430094847, −2.82033564212613923349790495533, −1.87638880867280198607018305083, 0, 1.87638880867280198607018305083, 2.82033564212613923349790495533, 4.00522230305247131920430094847, 4.46143013051310227158833590017, 5.80989038895996252692901571045, 6.66323789863247983935396491695, 7.46283883012578601583213147088, 8.095877094651885857593761101144, 9.201657130364874607788912355154

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