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 + 4·7-s + 9-s − 6·13-s − 15-s − 2·17-s + 4·19-s + 4·21-s − 8·23-s + 25-s + 27-s − 6·29-s − 4·35-s − 6·37-s − 6·39-s + 10·41-s − 4·43-s − 45-s + 8·47-s + 9·49-s − 2·51-s + 10·53-s + 4·57-s + 6·61-s + 4·63-s + 6·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.66·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s + 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.676·35-s − 0.986·37-s − 0.960·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s + 1.37·53-s + 0.529·57-s + 0.768·61-s + 0.503·63-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  =  $(2,\ 120,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.244910145$
$L(\frac12)$  $\approx$  $1.244910145$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
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 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−19.25531441542306, −18.11472715279224, −17.48597847685339, −16.28913908173465, −15.15548275690823, −14.53227586453705, −13.73753072310141, −12.27299498206974, −11.57665331369843, −10.34021526003760, −9.138788409882083, −7.929044973464792, −7.345943651768355, −5.288978885045043, −4.137288595248651, −2.191805633829980, 2.191805633829980, 4.137288595248651, 5.288978885045043, 7.345943651768355, 7.929044973464792, 9.138788409882083, 10.34021526003760, 11.57665331369843, 12.27299498206974, 13.73753072310141, 14.53227586453705, 15.15548275690823, 16.28913908173465, 17.48597847685339, 18.11472715279224, 19.25531441542306

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