Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

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

\[\begin{aligned} \Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{735} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 735,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−19.62313380590836, −18.93501661903010, −18.59174458557952, −17.75250881168677, −17.11758388779058, −16.09795172028608, −15.61969297730928, −14.74588211557607, −13.87125161959730, −13.16706216458194, −12.65229541857367, −11.33927396224944, −10.54178668489832, −9.948652838212899, −8.838960281218345, −8.413264856684473, −7.689939431823155, −6.741268487583156, −5.282943242472323, −4.364592268314344, −3.305931259647901, −1.843394199214780, 0, 1.843394199214780, 3.305931259647901, 4.364592268314344, 5.282943242472323, 6.741268487583156, 7.689939431823155, 8.413264856684473, 8.838960281218345, 9.948652838212899, 10.54178668489832, 11.33927396224944, 12.65229541857367, 13.16706216458194, 13.87125161959730, 14.74588211557607, 15.61969297730928, 16.09795172028608, 17.11758388779058, 17.75250881168677, 18.59174458557952, 18.93501661903010, 19.62313380590836

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