Properties

Degree 2
Conductor $ 3 \cdot 11 \cdot 19 $
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 − 2·4-s + 2·7-s + 9-s + 11-s − 2·12-s − 13-s + 4·16-s + 3·17-s + 19-s + 2·21-s + 6·23-s − 5·25-s + 27-s − 4·28-s + 8·31-s + 33-s − 2·36-s + 2·37-s − 39-s + 6·41-s + 8·43-s − 2·44-s − 6·47-s + 4·48-s − 3·49-s + 3·51-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s + 0.727·17-s + 0.229·19-s + 0.436·21-s + 1.25·23-s − 25-s + 0.192·27-s − 0.755·28-s + 1.43·31-s + 0.174·33-s − 1/3·36-s + 0.328·37-s − 0.160·39-s + 0.937·41-s + 1.21·43-s − 0.301·44-s − 0.875·47-s + 0.577·48-s − 3/7·49-s + 0.420·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(627\)    =    \(3 \cdot 11 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{627} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 627,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.631559008$
$L(\frac12)$  $\approx$  $1.631559008$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.50751231115631, −19.24499528679530, −18.34963244156326, −17.63063183690699, −17.12758377628094, −16.12671858719824, −15.04561407016022, −14.57634941033597, −13.87086883082605, −13.23108425481964, −12.32797783376588, −11.51061305183315, −10.38316548933612, −9.595676059271080, −8.885855868195722, −8.047058109228960, −7.389878449703635, −5.926151058893251, −4.883037272279540, −4.104972064748853, −2.893957436473753, −1.216786827465290, 1.216786827465290, 2.893957436473753, 4.104972064748853, 4.883037272279540, 5.926151058893251, 7.389878449703635, 8.047058109228960, 8.885855868195722, 9.595676059271080, 10.38316548933612, 11.51061305183315, 12.32797783376588, 13.23108425481964, 13.87086883082605, 14.57634941033597, 15.04561407016022, 16.12671858719824, 17.12758377628094, 17.63063183690699, 18.34963244156326, 19.24499528679530, 19.50751231115631

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