Properties

Degree 2
Conductor $ 2^{3} \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 + 6·13-s + 15-s − 6·17-s − 4·19-s + 25-s + 27-s − 2·29-s − 8·31-s − 4·33-s − 2·37-s + 6·39-s − 6·41-s + 12·43-s + 45-s + 8·47-s − 7·49-s − 6·51-s + 6·53-s − 4·55-s − 4·57-s + 12·59-s + 14·61-s + 6·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.960·39-s − 0.937·41-s + 1.82·43-s + 0.149·45-s + 1.16·47-s − 49-s − 0.840·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s + 1.79·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{120} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 120,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.269494277$
$L(\frac12)$  $\approx$  $1.269494277$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−18.98432119428583, −18.28979756477100, −17.47591383810055, −16.09223509041478, −15.52339803564940, −14.39728297449827, −13.23486091060546, −13.00221841062233, −11.15225413084276, −10.45550448318589, −9.042960162718977, −8.323057177226079, −6.892708107693131, −5.595851586326822, −3.953038348506774, −2.250024320040478, 2.250024320040478, 3.953038348506774, 5.595851586326822, 6.892708107693131, 8.323057177226079, 9.042960162718977, 10.45550448318589, 11.15225413084276, 13.00221841062233, 13.23486091060546, 14.39728297449827, 15.52339803564940, 16.09223509041478, 17.47591383810055, 18.28979756477100, 18.98432119428583

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