Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 \cdot 11 $
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 + 6-s − 7-s − 3·8-s + 9-s − 11-s − 12-s − 6·13-s − 14-s − 16-s − 2·17-s + 18-s + 4·19-s − 21-s − 22-s − 3·24-s − 6·26-s + 27-s + 28-s − 2·29-s + 8·31-s + 5·32-s − 33-s − 2·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.218·21-s − 0.213·22-s − 0.612·24-s − 1.17·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.883·32-s − 0.174·33-s − 0.342·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5775\)    =    \(3 \cdot 5^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5775} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5775,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.083259964$
$L(\frac12)$  $\approx$  $2.083259964$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−17.48566197043311, −17.05376819416841, −16.02231941408352, −15.58132066493719, −14.99452137840326, −14.35440324187024, −13.91565310322213, −13.41956067157130, −12.63349574702984, −12.32582539408050, −11.68232111763172, −10.66695217442116, −9.935008527873886, −9.398863350456849, −8.976172580570064, −7.995715080914812, −7.475061489247619, −6.666229192515516, −5.817684241623596, −5.031264896607378, −4.529408232945161, −3.686287903618178, −2.871141103608065, −2.295199291477628, −0.6441153764688498, 0.6441153764688498, 2.295199291477628, 2.871141103608065, 3.686287903618178, 4.529408232945161, 5.031264896607378, 5.817684241623596, 6.666229192515516, 7.475061489247619, 7.995715080914812, 8.976172580570064, 9.398863350456849, 9.935008527873886, 10.66695217442116, 11.68232111763172, 12.32582539408050, 12.63349574702984, 13.41956067157130, 13.91565310322213, 14.35440324187024, 14.99452137840326, 15.58132066493719, 16.02231941408352, 17.05376819416841, 17.48566197043311

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