Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 11 \cdot 19^{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  + 3-s + 2·5-s + 2·7-s + 9-s − 11-s + 2·13-s + 2·15-s + 4·17-s + 2·21-s − 25-s + 27-s + 8·29-s − 8·31-s − 33-s + 4·35-s − 10·37-s + 2·39-s − 8·41-s + 2·43-s + 2·45-s + 8·47-s − 3·49-s + 4·51-s + 2·53-s − 2·55-s + 12·59-s + 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.516·15-s + 0.970·17-s + 0.436·21-s − 1/5·25-s + 0.192·27-s + 1.48·29-s − 1.43·31-s − 0.174·33-s + 0.676·35-s − 1.64·37-s + 0.320·39-s − 1.24·41-s + 0.304·43-s + 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s − 0.269·55-s + 1.56·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(190608\)    =    \(2^{4} \cdot 3 \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{190608} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 190608,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.484633327$
$L(\frac12)$  $\approx$  $5.484633327$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 + T \)
19 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.11449189184167, −12.77385942768751, −12.19074964442227, −11.75569223524924, −11.17177831168911, −10.67106377389943, −10.11668017782655, −9.965664137481415, −9.304069752107180, −8.717980474800251, −8.411520313258384, −7.985037213075593, −7.365285007467369, −6.843726053355737, −6.416283169111813, −5.577605470924590, −5.334091518204978, −4.958433754426546, −3.985000888210626, −3.710366526033209, −3.024308611306244, −2.334204636444806, −1.888524321375392, −1.345147971494442, −0.6424837297628533, 0.6424837297628533, 1.345147971494442, 1.888524321375392, 2.334204636444806, 3.024308611306244, 3.710366526033209, 3.985000888210626, 4.958433754426546, 5.334091518204978, 5.577605470924590, 6.416283169111813, 6.843726053355737, 7.365285007467369, 7.985037213075593, 8.411520313258384, 8.717980474800251, 9.304069752107180, 9.965664137481415, 10.11668017782655, 10.67106377389943, 11.17177831168911, 11.75569223524924, 12.19074964442227, 12.77385942768751, 13.11449189184167

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