Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s + 4·11-s − 2·13-s + 15-s + 2·17-s − 4·19-s + 25-s + 27-s − 2·29-s + 4·33-s − 10·37-s − 2·39-s + 10·41-s − 4·43-s + 45-s − 8·47-s − 7·49-s + 2·51-s − 10·53-s + 4·55-s − 4·57-s + 4·59-s − 2·61-s − 2·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.696·33-s − 1.64·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s + 0.280·51-s − 1.37·53-s + 0.539·55-s − 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.248·65-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{240} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 240,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.596242222$
$L(\frac12)$  $\approx$  $1.596242222$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.55056720504788, −19.37149001108976, −18.20746879965587, −17.28295947719531, −16.68239544701663, −15.55824655031658, −14.52464393317952, −14.21566908678037, −13.03045565076762, −12.26787268519799, −11.16070118695891, −10.00510556501914, −9.265565204604990, −8.330811446905088, −7.123604782956306, −6.135875456050285, −4.688310528994096, −3.392743728100760, −1.823465130108694, 1.823465130108694, 3.392743728100760, 4.688310528994096, 6.135875456050285, 7.123604782956306, 8.330811446905088, 9.265565204604990, 10.00510556501914, 11.16070118695891, 12.26787268519799, 13.03045565076762, 14.21566908678037, 14.52464393317952, 15.55824655031658, 16.68239544701663, 17.28295947719531, 18.20746879965587, 19.37149001108976, 19.55056720504788

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