Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 $
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 − 4-s − 7-s − 3·8-s + 6·13-s − 14-s − 16-s + 2·17-s − 8·19-s + 8·23-s + 6·26-s + 28-s + 2·29-s + 4·31-s + 5·32-s + 2·34-s + 2·37-s − 8·38-s + 6·41-s − 4·43-s + 8·46-s + 8·47-s + 49-s − 6·52-s + 10·53-s + 3·56-s + 2·58-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s + 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s − 1.83·19-s + 1.66·23-s + 1.17·26-s + 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.883·32-s + 0.342·34-s + 0.328·37-s − 1.29·38-s + 0.937·41-s − 0.609·43-s + 1.17·46-s + 1.16·47-s + 1/7·49-s − 0.832·52-s + 1.37·53-s + 0.400·56-s + 0.262·58-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1575} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 1575,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.963829443$
$L(\frac12)$  $\approx$  $1.963829443$
$L(\frac{3}{2})$   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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−19.89773845597442, −19.01201171557312, −18.71812003148084, −17.96776849728957, −17.15086107735687, −16.62603491013510, −15.60791895264988, −15.17672750611422, −14.44405608315813, −13.65684048353028, −13.13775014985679, −12.66428112179456, −11.82803816445160, −10.90017490907870, −10.34594737512671, −9.161169131924908, −8.789607307934369, −7.977292875613883, −6.635855619990924, −6.161498788632945, −5.260033875974153, −4.288991635446869, −3.632961218703625, −2.647659800941371, −0.9170375745596437, 0.9170375745596437, 2.647659800941371, 3.632961218703625, 4.288991635446869, 5.260033875974153, 6.161498788632945, 6.635855619990924, 7.977292875613883, 8.789607307934369, 9.161169131924908, 10.34594737512671, 10.90017490907870, 11.82803816445160, 12.66428112179456, 13.13775014985679, 13.65684048353028, 14.44405608315813, 15.17672750611422, 15.60791895264988, 16.62603491013510, 17.15086107735687, 17.96776849728957, 18.71812003148084, 19.01201171557312, 19.89773845597442

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