Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 47^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s + 4·11-s + 12-s + 2·13-s + 15-s − 16-s + 2·17-s − 18-s − 4·19-s + 20-s − 4·22-s − 3·24-s + 25-s − 2·26-s − 27-s + 2·29-s − 30-s − 5·32-s − 4·33-s − 2·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.852·22-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.883·32-s − 0.696·33-s − 0.342·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 33135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 33135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(33135\)    =    \(3 \cdot 5 \cdot 47^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{33135} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 33135,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4656717163$
$L(\frac12)$  $\approx$  $0.4656717163$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;47\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;47\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
47 \( 1 \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
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

−14.96988121065474, −14.56049920496281, −13.93829422224649, −13.42617754646617, −12.90386588182312, −12.13087947031564, −11.93232508554282, −11.22957095901129, −10.58963813868746, −10.32762458061877, −9.588327639083278, −9.079182332929625, −8.525631496212021, −8.159608564829133, −7.410659396144726, −6.774486557591058, −6.358591877837365, −5.589314827332597, −4.802636589904197, −4.425642990265391, −3.677157703038847, −3.184899025362603, −1.631770905444037, −1.466212756471995, −0.3219951926837716, 0.3219951926837716, 1.466212756471995, 1.631770905444037, 3.184899025362603, 3.677157703038847, 4.425642990265391, 4.802636589904197, 5.589314827332597, 6.358591877837365, 6.774486557591058, 7.410659396144726, 8.159608564829133, 8.525631496212021, 9.079182332929625, 9.588327639083278, 10.32762458061877, 10.58963813868746, 11.22957095901129, 11.93232508554282, 12.13087947031564, 12.90386588182312, 13.42617754646617, 13.93829422224649, 14.56049920496281, 14.96988121065474

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