Properties

Label 2-1680-1.1-c1-0-17
Degree $2$
Conductor $1680$
Sign $-1$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
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 − 6·13-s − 15-s + 2·17-s + 8·19-s + 21-s − 8·23-s + 25-s − 27-s − 2·29-s − 4·31-s − 35-s − 2·37-s + 6·39-s − 6·41-s − 4·43-s + 45-s − 8·47-s + 49-s − 2·51-s + 10·53-s − 8·57-s − 4·59-s − 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.258·15-s + 0.485·17-s + 1.83·19-s + 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.169·35-s − 0.328·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 1.37·53-s − 1.05·57-s − 0.520·59-s − 0.256·61-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: yes
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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−9.242967464128379637809586382976, −7.975429118345566696857391954617, −7.31447554122255167880309018857, −6.55815443374401606221787868995, −5.48396360051002103094444371014, −5.15942873071656623278244316377, −3.89765231085728538463298179402, −2.82363836469554907617267416973, −1.63265803663490016612214771155, 0, 1.63265803663490016612214771155, 2.82363836469554907617267416973, 3.89765231085728538463298179402, 5.15942873071656623278244316377, 5.48396360051002103094444371014, 6.55815443374401606221787868995, 7.31447554122255167880309018857, 7.975429118345566696857391954617, 9.242967464128379637809586382976

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