Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 19^{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 − 6-s + 2·7-s + 8-s + 9-s − 11-s − 12-s + 4·13-s + 2·14-s + 16-s − 6·17-s + 18-s − 2·21-s − 22-s + 6·23-s − 24-s − 5·25-s + 4·26-s − 27-s + 2·28-s − 6·29-s − 8·31-s + 32-s + 33-s − 6·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.436·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s − 25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.174·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23826 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23826 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(23826\)    =    \(2 \cdot 3 \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{23826} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 23826,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 + T \)
19 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.64826274202153, −15.16428600823998, −14.65608877277109, −14.10672571576464, −13.31846806887349, −13.00797045473693, −12.77862407740910, −11.62576183091605, −11.32903579775606, −11.12344328669381, −10.54235244629621, −9.706134848702614, −9.041959744139241, −8.486678739985864, −7.730979528687840, −7.228693914534644, −6.549598060669407, −5.962229192507064, −5.415131751642721, −4.846928782237787, −4.149368738963716, −3.703395210121601, −2.700261302739469, −1.910265346656847, −1.252279047260178, 0, 1.252279047260178, 1.910265346656847, 2.700261302739469, 3.703395210121601, 4.149368738963716, 4.846928782237787, 5.415131751642721, 5.962229192507064, 6.549598060669407, 7.228693914534644, 7.730979528687840, 8.486678739985864, 9.041959744139241, 9.706134848702614, 10.54235244629621, 11.12344328669381, 11.32903579775606, 11.62576183091605, 12.77862407740910, 13.00797045473693, 13.31846806887349, 14.10672571576464, 14.65608877277109, 15.16428600823998, 15.64826274202153

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