Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Sign $-1$
Motivic weight 1
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

\( d \)  =  \(2\)
\( N \)  =  \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1680} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1680,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.55407288103342, −18.52284586388433, −18.12915602675311, −17.47168242572083, −16.64981033265031, −16.38150726977025, −15.49216110742040, −14.71646032622622, −14.01114437290045, −13.40955492181118, −12.40727267653875, −12.05265024400696, −11.35809411677126, −10.13723594391339, −9.937991188085223, −9.242967464128380, −7.975429118345567, −7.314475541222552, −6.558154433744016, −5.483963600510021, −5.159428730716566, −3.897652310857285, −2.823638364695549, −1.632658036634900, 0, 1.632658036634900, 2.823638364695549, 3.897652310857285, 5.159428730716566, 5.483963600510021, 6.558154433744016, 7.314475541222552, 7.975429118345567, 9.242967464128380, 9.937991188085223, 10.13723594391339, 11.35809411677126, 12.05265024400696, 12.40727267653875, 13.40955492181118, 14.01114437290045, 14.71646032622622, 15.49216110742040, 16.38150726977025, 16.64981033265031, 17.47168242572083, 18.12915602675311, 18.52284586388433, 19.55407288103342

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